SOME EXTENSIONS OF GRAPH SATURATION TO EDGE COLORED,
ORIENTED, AND SUBDIVIDED GRAPHS
by
Craig M. Tennenhouse
Master of Arts, University of Colorado, 2001
A thesis submitted to the
University of Colorado Denver
in partial fulfillment
of the requirements for the degree of
Doctor of Philosophy
Applied Mathematics
2010
This thesis for the Doctor of Philosophy
degree by
Craig M. Tennenhouse
has been approved
by
Michael S. Jacobson
Richard Lundgren
Michael Ferrara
Ellen Gethner
Jason Williford
Date
Tennenhouse, Craig M. (Ph.D., Applied Mathematics)
Some Extensions of Graph Saturation to Edge Colored, Oriented, and Subdivided Graphs
Thesis directed by Professor Michael S. Jacobson
ABSTRACT
Extremal Graph Theory is the particular subfield of Graph Theory concerned with maximizing and minimizing certain parameters associated with
graphs and digraphs. Beginning with Mantel [46] and Ramsey [52], the study
of extremal graphs was extended by Turán [56] into edge-maximum clique free
graphs. Similar problems, particularly those in this thesis, are therefore referred
to as Turán type problems.
Erdös, Hajnal, and Moon [21] generalized Turán’s result even further, initiating the study of graph saturation. A graph H is G-free if it contains no
subgraph isomorphic to G. H is G-saturated if it is edge-maximally G-free.
Similarly, given a family F of graphs, H is F-saturated if it is G-free for every
graph G in F but the addition of any edge from the complement of H creates some graph in F. Erdös, Hajnal, and Moon characterized, in particular,
edge-minimum edge-maximally clique free graphs on a fixed number of vertices.
Various others have characterized smallest edge-maximally G-free graphs for
other classes of graphs G. In this thesis the concept of saturation is extended
to oriented graphs, characterizations of extremal interval graphs and interval
bigraphs are offered, minimal F-saturated graphs are determined where F is a
family of subdivided graphs, a new saturation parameter on edge-colored graphs
is introduced, and impartial and partizan games related to oriented graph saturation are examined.
This abstract accurately represents the content of the candidate’s thesis. I
recommend its publication.
Signed
Michael S. Jacobson
DEDICATION
This thesis is dedicated to my father Don, who always told me to do good, and
my daughters Ceili and Hadassah, who give me a reason to.
ACKNOWLEDGMENT
I would first like to thank my adviser, Michael S. Jacobson, for guiding me
from being interested in mathematics to actually engaging in mathematics. I
would also like to give credit to the collaborators with whom I’ve worked on this
and other material over the past few years, and have mentioned in the following
pages. Lastly, and certainly most importantly, I thank my wife, Laura, for her
infinite patience, love, and support. I could not have accomplished this without
her.
CONTENTS
Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
ix
1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1
1.1
Prologue . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1
1.2
Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2
1.3
Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6
2. Oriented Graph Saturation . . . . . . . . . . . . . . . . . . . . . . . .
13
2.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
13
2.2
Motivation for extension to oriented graphs . . . . . . . . . . . . .
13
2.3
Existence of saturated oriented graphs . . . . . . . . . . . . . . . .
14
2.4
Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
18
2.5
Some Upper Bounds . . . . . . . . . . . . . . . . . . . . . . . . . .
24
2.6
The minimum order of a D-saturated oriented graph . . . . . . . .
27
2.7
Future directions . . . . . . . . . . . . . . . . . . . . . . . . . . . .
29
3. Maximal Non-Interval Graphs and Bigraphs . . . . . . . . . . . . . .
30
3.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
3.2
Edge-Maximal Non-Interval Graphs . . . . . . . . . . . . . . . . . .
31
3.3
Edge-Maximal Non-Interval Bigraphs . . . . . . . . . . . . . . . . .
33
3.4
Edge-Maximal Split Non-Interval Graphs . . . . . . . . . . . . . . .
43
3.5
Other Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
45
4. Subdivided Graph Saturation . . . . . . . . . . . . . . . . . . . . . .
48
vii
4.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
48
4.2
Cycles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
49
4.2.1
History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
49
4.2.2
t≤6. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
49
4.2.3
General Bounds . . . . . . . . . . . . . . . . . . . . . . . . . . . .
64
Complete Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . .
76
5. Edge-Critical G, H Colorings . . . . . . . . . . . . . . . . . . . . . . .
85
5.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
85
5.2
Kneser Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
86
5.3
Some Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
89
6. Conclusion and Futher Directions . . . . . . . . . . . . . . . . . . . .
96
6.1
Oriented Graph Saturation . . . . . . . . . . . . . . . . . . . . . . .
96
6.2
Maximal Non-Interval Graphs and Bigraphs . . . . . . . . . . . . .
98
6.3
Subdivided graph saturation . . . . . . . . . . . . . . . . . . . . . .
99
6.4
Edge-Critical G, H Colorings . . . . . . . . . . . . . . . . . . . . .
100
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
102
4.3
viii
FIGURES
Figure
1.1
The unique graph realizing sat(8, K6 ) . . . . . . . . . . . . . . . . .
8
1.2
The unique smallest 3K3 -saturated graph on 12 vertices . . . . . .
8
1.3
The unique smallest K1,8 -saturated graph on 14 vertices . . . . . .
9
1.4
A P6 saturated graph . . . . . . . . . . . . . . . . . . . . . . . . . .
10
2.1
G with edges xy and xz, G0 with edge x0 z 0 and non-edge x0 y 0 , and G∗
with edge x∗ z ∗ and non-edge x∗ y ∗ . . . . . . . . . . . . . . . . . . .
14
2.2
The oriented graph HG,a . . . . . . . . . . . . . . . . . . . . . . . .
15
2.3
Cut vertices v 0 , v ∗ and non-cut vertices w0 , w∗ in HG,a − F . . . . .
16
2.4
The union of HG,a with isolated vertices . . . . . . . . . . . . . . .
19
2.5
The smallest P~3 -saturated oriented graph on 7 vertices . . . . . . .
19
2.6
A P~5 -saturated oriented graph . . . . . . . . . . . . . . . . . . . . .
20
2.7
A K1,(3,3) -saturated oriented graph . . . . . . . . . . . . . . . . . .
23
2.8
A transitive tournament saturated oriented graph, consisting of a
transitive tournament joined from a set of isolated vertices . . . . .
24
A P~m -saturated oriented graph . . . . . . . . . . . . . . . . . . . .
26
2.10 No oriented graph on 3 vertices is C3 -saturated . . . . . . . . . . .
28
2.9
2.11 The addition of either arc v1 vm or arc vm v1 generates a hamiltonian
cycle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
28
3.1
The 3-sun in which the white vertices form an AT . . . . . . . . . .
32
3.2
The bold edges form an asteriodal triple of edges . . . . . . . . . .
34
ix
3.3
Three graphs, called insects, that contain an exobiclique . . . . . .
3.4
An edge asteroid of order 5. Note that joining either vertex in e4
34
to any vertex on the path between the two white vertices, inclusive,
eliminates this property. . . . . . . . . . . . . . . . . . . . . . . . .
35
3.5
An ATE that is not an EA of order 3 . . . . . . . . . . . . . . . . .
35
3.6
The addition of edge {xi+1 , yi+k+1 } creates an ATE . . . . . . . . .
40
3.7
Possible configurations of the vertices ai , ai+1 . . . . . . . . . . . .
41
3.8
ai , ai+1 in the same partite set . . . . . . . . . . . . . . . . . . . . .
41
3.9
Possible configurations when ai , ai+1 in different partite sets . . . .
42
3.10 The forbidden subgraphs G1 and G3 . . . . . . . . . . . . . . . . .
43
3.11 The basic minimally non-circular arc graphs . . . . . . . . . . . . .
46
3.12 A graph and its circular arc representation . . . . . . . . . . . . . .
47
4.1
The Coxeter graph . . . . . . . . . . . . . . . . . . . . . . . . . . .
51
4.2
A (1, 3)-tree and the associated graph in T18 . . . . . . . . . . . . .
51
4.3
S(C5 ) builders (subscripts indicate value of n (mod 7)) . . . . . . .
56
4.4
Exceptional K4 -type Lobes . . . . . . . . . . . . . . . . . . . . . .
59
4.5
The graphs E3 , E30 , and E300
. . . . . . . . . . . . . . . . . . . . . .
61
4.6
A graph G and the modifications G(e) and G(v, e) . . . . . . . . .
67
4.7
The graph J3 and a C12 -builder based on it . . . . . . . . . . . . .
68
4.8
Every edge in a modified snark is on a long cycle . . . . . . . . . .
71
4.9
An S(K5 )-saturated graph of order 18. (d = 4, r = 2) . . . . . . . .
79
4.10 Examples of K5 -subdivisions in G(5, 18)+uv. Open circles represent
the branch vertices of the subdivisions formed. . . . . . . . . . . . .
81
4.11 G(6, 21), an S(K6 )-saturated graph of order 21. (d = 4, r = 1) . . .
82
x
4.12 Examples of K6 -subdivisions in G(6, 21)+uv. Open circles represent
the branch vertices of the subdivisions formed. . . . . . . . . . . . .
83
5.1
An edge-critical P3 , P3 coloring of a graph . . . . . . . . . . . . . .
86
5.2
The Petersen Graph KG(5, 2) . . . . . . . . . . . . . . . . . . . . .
87
5.3
A colored subgraph of every edge-critical K3 , K3 coloring of a graph
90
5.4
Minimal edge-critical P3 , K6 coloring of F . . . . . . . . . . . . . .
91
5.5
An edge-critical G, P3 coloring of a graph H in which n(G) =
7, δ(G) = 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
92
5.6
An edge-critical G, G coloring for κ0 (G) = 1 . . . . . . . . . . . . .
93
5.7
G has a pair of adjacent degree 2 vertices not on a triangle . . . . .
94
6.1
The addition of arc xy to s(K4 ∪ K1 ) does not create a K1,(4,0) . . .
98
xi
1. Introduction
1.1 Prologue
Graph Saturation is a subfield of Extremal Graph Theory, which itself has
its roots in Ramsey Theory [52] in 1930 and the work of Mantel [46] in 1907.
Brought to the forefront of Graph Theory by Turán [56] in 1954, Graph Saturation has been popularized and extended by a number of mathematicians
throughout the latter half of the twentieth century. G-free graphs, that is
graphs not containing subgraphs isomorphic to some simple graph G, specifically edge-maximal G-free graphs, have been studied in particular with respect
to maximizing and minimizing number of edges. In this chapter we give a brief
history of the study of Graph Saturation and introduce definitions and notation
that will prove useful throughout.
One mostly overlooked problem within Graph Saturation is the extension
of the field into oriented graphs. In Chapter 2 we attempt to address this
missing piece of the puzzle by demonstrating the existence of arc-maximal Dfree graphs for all oriented graphs D, and determine the minimum number of
edges in specific cases.
Interval graphs were first introduced in 1957 with the work of Hajos [33].
Their applications are myriad, with current interest driven primarily by the
study of genomics. Eckhoff considered interval graphs in light of Extremal Graph
Theory in [19]. In Chapter 3 we change our focus from maximizing and minimizing the number of edges in saturated graphs and oriented graphs to continue
1
this work. We attempt to characterize all edge-maximal graphs meeting certain
properties related to interval graphs.
In Chapter 4 we return to minimizing the number of edges in certain saturated graphs. Topological minors, related to graph subdivisions, are useful in
determining planarity [42] and network design. We give upper bounds for the
number of edges in certain saturated graphs related to subdivisions, and provide
an asymptotic result.
An edge-coloring approach to Graph Saturation has also become an area
of interest with so-called Ramsey-Turán Theory. Hanson and Toft [34] and
Grossman [31] have made inroads into this area. In Chapter 5 we look at a new
problem in this area and determine some results. In the final chapter we extend
some combinatorial games on graphs to oriented graphs.
1.2 Notation
We begin with some terminology and definitions, and refer to [58] as an
excellent resource.
A graph G = (V, E) is a non-empty finite set V (G) of vertices together with
a set E(G) of unordered pairs of distinct vertices of G called edges. For a graph
G with vertices u, v ∈ V (G) the edge e = {u, v}, or sometimes uv, is said to join
u and v. For brevity we will sometimes use the notation v ∈ G (e ∈ G) instead
of v ∈ V (G) (e ∈ E(G)). We refer to u and v as being adjacent, and denote this
relationship by u ∼ v. If u and v are not adjacent, this will be denoted u v
and we will sometimes refer to a non-adjacency between the vertices. A graph
defined in this way, with at most one edge between any pair of vertices and no
vertex v with the property that v ∼ v, is called a simple graph. Throughout we
2
primarily consider simple graphs and define any variation explicitly.
The order of a graph G is the cardinality of its vertex set, denoted either
|V (G)| or n(G). The size of G is the cardinality of its edge set, denoted either
|E(G)| or e(G). and For a vertex v ∈ V (G), the degree of v, denoted dG (v)
or simply d(v) where the context makes clear the graph G, is the number of
edges incident with v. The neighborhood of v, denoted NG (v) or N (v), is the set
of vertices adjacent to v. The minimum degree and maximum degree of G are
denoted δ(G) and ∆(G), respectively. A graph in which δ(G) = r = ∆(G) for
some integer r is referred to as r-regular.
A graph H is a subgraph of G if V (H) ⊆ V (G) and E(H) ⊆ E(G), and here
G is referred to as a supergraph of H. For simplicity this is denoted H ⊆ G. For
a graph G and a set S ⊆ V (G) we define the induced subgraph G[S] to be the
graph with vertex set S and edge set T = {e : e ∈ E(G), e ⊆ S}. Note that by
these definitions not all subgraphs are induced subgraphs of G.
If G and H are graphs then their union is the graph G ∪ H with vertex set
V (G) ∪ V (H) and edge set E(G) ∪ E(H). If v is a vertex not in G then by
G + v we will mean the graph G ∪ {v}. The complement of G, denoted Gc , is
the unique graph with same same vertex set as G and an edge between vertices
u and v if an only if u v in G. If e ∈ Gc then G + e will denote the graph
with vertex set V (G) and edge set E(G) ∪ {e}. Similarly, if v and e are in V (G)
and E(G) respectively then G − v, G − e denote, respectively, the graph G with
v and all incident edges removed and the graph G with edge e removed.
A graph G in which there exists a partition of V (G) into sets X and Y
where there is no edge among the vertices in X nor among the vertices in Y is
3
called bipartite, or simply a bigraph, with parts X and Y . We will denote this
by G = (X, Y ). If V (G) can be partitioned into k ≥ 2 such sets then G is called
k-partite, and more generally multipartite.
Let G and H be graphs. G is isomorphic to H if there is a bijection φ
between their respective vertex sets that retains adjacencies and non-adjacencies,
which we will denote G ∼
= H. The join of G and H, denoted G ∨ H, is the graph
with vertex set V (G) ∪ V (H) and edge set consisting of all edges in E(G) and
E(H), as well as all edges of the form {u, v} where u ∈ G, v ∈ H.
A clique in a graph G is a subgraph H ⊆ G in which there is an edge between
every pair of vertices in H. If G is itself a clique then G is a complete graph
of order n(G), or G ∼
= Kn(G) . An independent set in G is the complement of a
clique. A bipartite graph G = (X, Y ) is complete if for every x ∈ X, y ∈ Y G
contains the edge {x, y}. If |X| = s, |Y | = t then the complete bipartite graph
with parts X and Y is denoted Ks,t .
If u and v are vertices of the graph G then a u-v path, if it exists, is an ordered
set of distinct vertices ux1 x2 . . . xk v in which {u, x1 }, {xk , v}, {xi , xi+1 }k−1
i=1 ∈
E(G). The length of this path is the number of edges traversed, i.e. one less
than the number of vertices, inclusive. We will use the notation Pk for a path
of length k − 1. If P = v1 , v2 , . . . , vk and P1 = v2 , . . . , vk−1 then we will say that
P = v1 P1 vk . The distance between u and v is the length of the shortest path
over all u − v paths. A cycle is a path with the same starting and ending vertex.
Its length is defined to be the number of vertices. If a cycle C has length k then
it is sometimes referred to as a k-cycle, and will be denoted Ck . G is connected if
for every pair of vertices u, v ∈ G, G contains a u − v path. If G is disconnected
4
then its maximal connected subgraphs are referred to as its components. A tree
is a connected graph containing no cycles. A forest is a graph in which all of its
components are trees.
Let G and H be graphs. H is G-free if it does not contain a subgraph
isomorphic to G. H is maximally G-free, or G-saturated, if H is G-free but
for any edge e ∈ H c the graph H + e contains a subgraph isomorphic to G.
For a fixed n, the saturation number sat(n, G) denotes the fewest edges in a
G-saturated graph on n vertices, and the extremal number ex(n, G) denotes
the greatest. Finally, an edge e in G incident with the vertices u, v ∈ G can
be subdivided by the removal of e, the addition of the vertex x and the edges
{x, u}, {x, v}. If an edge of G is subdivided then G is also said to be subdivided.
We now turn our attention to oriented graphs. An orientated graph D =
(V, A) consists of a set of vertices V (D) and a set of arcs A(D), consisting of
ordered pairs of vertices in V (D). For vertices u, v ∈ D the arc (u, v) has initial
vertex u and terminal vertex v. If D is the oriented graph D = (V, A) and G
is the graph with V (G) = V (D) and E(G) = {{u, v} : (u, v) ∈ A(D)} then G
is the underlying graph of D and D is an orientation of G. Subdigraphs and
superdigraphs are defined similarly to subgraphs and supergraphs. An oriented
graph is (r, r)-regular if each vertex has precisely r out-neighbors and r inneighbors.
An ordered set of vertices ux1 x2 . . . xk v in which (u, x1 ), (xi , xi+1 )k−1
i=1 , (xk , v) ∈
A(D) is a directed u-v path. If u = v then this is a directed cycle. A tournament
T is an orientation of a complete graph. If there is a u − v directed path for
every ordered pair of vertices u, v ∈ T then T is strong. If there is a labeling of
5
the vertices of T of the form {v1 , v2 , . . . , vn } such that for every 1 ≤ i ≤ (n − 1)
(vi , vi+1 ) ∈ A(T ) then T is the transitive tournament of order n.
1.3 Background
The first theorem addressing edge-extremal graphs with forbidden subgraphs
was the following result of Mantel and his students in 1907 [46].
Theorem 1.3.1. Let G be a simple graph on n vertices. If G has more than
2
b n4 c edges then G contains a subgraph isomorphic to K3 .
2
Mantel proved that this result is best possible, thus ex(n, K3 ) = b n4 c.
Extremal Graph Theory continued with a theorem of Ramsey’s from the
field of philosophy. It was originally stated as follows [52]:
Theorem 1.3.2. Let Γ be an infinite class, and µ and γ positive integers; and
let all γ-combinations of the members of Γ be divided in any manner into µ
mutually exclusive classes Ci (i = 1, 2, . . . , µ), so that every γ-combination is a
member of one and only one Ci ; then assuming the axiom of selections, Γ must
contain an infinite sub-class ∆ such that all the γ-combinations of the members
of ∆ belong to the same Ci .
Theorem 1.3.2 is often restated using only graph theoretic terminology:
Theorem 1.3.3. Let a, b ≥ 1 be integers. There exists a smallest integer r =
R(a, b) such that every coloring of the edges of the graph Kr with the colors red
and blue contains either a red subgraph isomorphic to Ka or a blue subgraph
isomorphic to Kb .
6
Turán extended Theorem 1.3.1 to larger forbidden complete graphs in [56].
Note that it is often convenient and more elucidating to discuss graphs that
realize saturation numbers rather than the values themselves.
Definition 1.3.4. Let r, n be integers, 1 < r ≤ n. Denote by T r−1 (n) the
unique complete (r − 1)-partite graph on n vertices the order of whose partition
sets differ by at most 1, and by tr−1 (n) the number of edges in T r−1 (n). This
family comprises the Turán graphs.
Theorem 1.3.5. [56] Let G be a simple Kr -free graph. Then, G has at most
tr−1 (n) edges, and G is isomorphic to T r−1 (n).
Therefore, ex(n, Kr ) = tr−1 (n). We refer to similar problems as Turán-type
problems.
Erdös, Hajnal, and Moon [21] extended this idea in the other direction, and
determined the fewest number of edges in a Kr -saturated graph on n vertices,
and proved that their construction is unique, (Fig. 1.1).
Theorem 1.3.6. The smallest number of edges in a Kr -saturated graph on n
vertices is r−1
+ (r − 2)(n − r + 2). The only Kr -saturated graph on n vertices
2
containing sat(n, Kr ) edges is Kr−2 ∨ K̄n−r+2 .
Thus, the saturation number of Kr on n vertices is sat(n, Kr ) =
r−1
2
+ (r −
2)(n − r + 2).
We continue our generalization of complete graphs even further by considering disjoint unions of cliques.
Let t be a positive integer and G a graph. By tG we mean t disjoint copies
of G. In [24] Faudree, Ferrara, Gould, and Jacobson determined precise values
7
Figure 1.1: The unique graph realizing sat(8, K6 )
Figure 1.2: The unique smallest 3K3 -saturated graph on 12 vertices
for both sat(n, tKp ) and sat(n, Kp ∪ Kq ).
Theorem 1.3.7. Let t, p, n be positive integers, p > 1. sat(n, tKp ) is realized
by the graph Kp−2 ∨ {(t − 1)Kp+1 ∪ K̄n−pt−t+3 }.
Though it is not yet known whether or not this graph uniquely realizes
sat(n, tKp ) for all t, it does when t = 2 or t = 3, (Fig. 1.2).
Theorem 1.3.8. Let 1 < p ≤ q, n be integers. sat(n, Kp ∪ Kq ) is realized by the
graph Kp−2 ∨ {Kq+1 ∪ Kn−q−p+1 }.
We now turn our attention to paths, trees, and forests.
Kaszonyi and Tuza [41] determined the saturation number for various acyclic
graphs.
8
Figure 1.3: The unique smallest K1,8 -saturated graph on 14 vertices
Theorem 1.3.9.
sat(n, K1,k−1 ) =



k−1
2
+

 d k−2 n −
2
n−k+1
2
if k ≤ n ≤
(k−1)2
e
8
if
3k−3
2
3k−3
2
≤n
The graphs that realize these values are Kk−1 ∪Kn−k+1 and the disjoint union
of Kb k c with a (k − 1)-regular graph on n − b k2 c vertices (Fig. 1.3), respectively,
2
and are unique.
In [41], Kasonyi and Tuza show that for any non-star tree T on k vertices,
sat(n, T ) < sat(n, K1,k−1 ). This is especially evident in the case where T is a
star with a single subdivided edge. In fact, in [24] the authors show that, if T0
is the subdivided star on k vertices and T is any tree isomorphic to neither T0
nor K1,k−1 , then n − b (n+k−2)
c = sat(n, T0 ) < sat(n, T ) < sat(n, K1,k ).
k
This brings us to a very curious property of the saturation function. It is
well known that for graphs G ⊆ H and an integer n, ex(n, G) ≤ ex(n, H). It is
also clear that for families F ⊆ R of graphs, ex(n, R) ≤ ex(n, F). The same is
not true for the saturation function.
To show the non-monotonicity of sat in the former case, consider the graph
Gk = K1,k−1 + e, that is, a star with the addition of an edge between two
leaves. Pikhurko showed [51] that sat(n, Gk ) = n − 1 < sat(n, K1,k−1 ). In fact,
9
Figure 1.4: A P6 saturated graph
it is K1,n−1 itself that realizes this value. In the latter case, the authors of [23]
consider F = {Gk }, R = {Gk , K1,k−1 }. Again, it is clear that a star on n vertices
realizes sat(n, F) = n − 1 and that sat(n, R) = sat(n, K1,k−1 ), which we see in
Theorem 1.3.9 is larger than n − 1.
Theorem 1.3.10.
Let ak =


 3 · 2m−1 − 2 if k = 2m

 4 · 2m−1 − 2 if k = 2m + 1
Then, for n ≥ ak , sat(n, Pk ) = n − b anm c and is uniquely realized by a forest
with b ank c components. Each component is a tree with depth at most d m2 e and
is 3-regular except for the leaves and at most one vertex, of degree 4 or more,
which is adjacent to a leaf, (Fig. 1.4).
It is natural to consider the extreme case in which we are looking for a path
of order n on n vertices, i.e. a hamiltonian path. In [28] Frick and Singleton
determined sat(n, Pn ) for large n.
Theorem 1.3.11. For n ≥ 54 sat(n, Pn ) = d 3n−2
e.
2
An m-path cover of a graph G is is a subgraph H consisting of at most m
disjoint paths with the property that V (H) = V (G). In [22] Dudek, Katona,
10
and Wojda generalized the result from Theorem 1.3.11 to m-path covers. We
can think of this as a sort of linear forest saturation.
Theorem 1.3.12. Let 1 < m ≤ n be integers, and let mP CSn denote the family
of graphs on n vertices containing an m-path cover. Then,
3
n
2
− 3m − 3 ≤
sat(n, mP CSn ) ≤ 32 n − 2m + 2.
Now we consider Ct -saturated graphs. We will only examine the few exact
known results in this chapter. A more extensive list of results, including bounds
for a number of cases, can be found in the introduction to the fourth chapter of
this thesis.
Note that, since C3 = K3 , the value for sat(n, C3 ) is determined by Theorem
1.3.6. In 1972, Ollmann [48] and later Tuza [57] determined the saturation
number for cycles of length 4, and determined all graphs realizing this value.
Theorem 1.3.13. Let n ≥ 5. sat(n, C4 ) = d 3n−5
e.
2
In 1995 Fisher, Fraughnaugh, and Langley [26] found an upper bound for
sat(n, C5 ), but it was not until 2009 when Y.-C. Chen [9] showed this to be a
lower bound as well, for sufficiently large n.
Theorem 1.3.14. sat(n, C5 ) ≤ d 10
(n − 1)e. This becomes equality for n ≥ 21.
7
Other authors have examined saturated graphs in a variety of other contexts. Hajnal [32] and Duffus-Hanson [18] considered the result of placing degree restrictions on clique saturated graphs. Hanson and Toft [34] related graph
saturation to edge-coloring, which we will look into further in chapter 5. Finally,
Bollobás [3] and Pikhurko [49], [50] studied saturated hypergraphs. See [23] by
11
J. Faudree, R. Faudree, and J. Schmitt for an excellent survey on the topic,
including a number of other results.
12
2. Oriented Graph Saturation
2.1 Introduction
Throughout this chapter we will refer to a simple graph as a graph. Let a(n
oriented) graph H contain G as a subgraph (subdigraph). We call the subgraph
of H that is isomorphic to G an embedding of G in H. Recall that for any
n ≥ n(G) there is a G-saturated graph H on n vertices, so both sat(n, G) and
ex(n, G) are well-defined. Let D, F be oriented graphs such that D is not a
subdigraph of F but the addition of any arc between nonadjacent vertices of F
results in a copy of D. We say that F is D-saturated. If G is a graph and D is
an orientation of G then G = u(D), the underlying graph of D.
2.2 Motivation for extension to oriented graphs
While a number of results have been obtained regarding simple graph saturation, there has been very little work on oriented graph saturation. The little
that has been determined has been restricted to orientations of multigraphs [8],
[20].
Arc-extremal oriented graphs with forbidden subdigraphs are a natural extension of edge-extremal graphs with forbidden subgraphs. However determining
the existence of D-saturated oriented graphs is not as trivial as in the undirected
case. To see this, we need only consider D containing a directed cycle. If F
is an acyclic oriented graph with non-adjacent vertices u, v then an arc can be
added between them, with the resulting superdigraph acyclic.
This led to a second problem. Given a simple graph G and integer n ≥ n(G),
ex(n, G) has traditionally been defined to be the maximum number of edges in
13
z
x
y
z'
y*
x'
y'
x*
z*
Figure 2.1: G with edges xy and xz, G0 with edge x0 z 0 and non-edge x0 y 0 , and
G∗ with edge x∗ z ∗ and non-edge x∗ y ∗
a G-free simple graph on n vertices. Considering the transitive tournament
on n vertices, a strict extension of this definition to arcs in oriented simple
graphs would mean that ex(n, D) = n2 for any oriented graph with directed
cycles. This clearly does not reflect the spirit of the study of extremal graphs and
oriented graphs. In order to extend the definition to something more meaningful,
we denote by ex(n, D) the maximum number of arcs in a D-saturated oriented
simple graph on n vertices.
2.3 Existence of saturated oriented graphs
Consider an undirected graph G. For x, y ∈ V (G) we call a = xy ∈
/ E(G) a
non-edge. Define the surgery HG,a in the following way:
Let G0 and G∗ be two copies of the graph G − xy, and for every vertex
v ∈ V (G) label its associated vertices v 0 ∈ G0 , v ∗ ∈ G∗ . Similarly, for every edge
α ∈ E(G) − {a} label the associated edges α0 ∈ G0 , α∗ ∈ G∗ , (see Figure 2.1). If
14
z'
x'=y*
y'=x*
z*
Figure 2.2: The oriented graph HG,a
D is an oriented graph with arc a then define HD,a analogously.
Let HG,a be the graph obtained by identifying x0 with y ∗ and x∗ with y 0
(see Figure 2.2). We will show that the graph HG,a does not contain a subgraph
isomorphic to G via induction on the order of G.
Lemma 2.3.1. If v is a cut vertex of a graph G that is not an endpoint of the
edge a, F is a connected subgraph of HG,a , and neither v 0 nor v ∗ are in F , then
there is another vertex w ∈ G that is not an endpoint of the edge a such that
neither w0 nor w∗ are in F and w is not a cut vertex of G.
Proof. Let A, B be components of G − v such that a ∈ A. The component B
contains at least one vertex w that is not a cut vertex of G. Every path from the
endpoints of the edge a to w in G contains v, so if the embedding of F contains
w0 or w∗ then it must also contain either v 0 or v ∗ . If neither v 0 nor v ∗ are in the
embedding of F , then neither are w0 nor w∗ , (see Figure 2.3).
Theorem 2.3.2. If G is a connected graph of order at least two, then for any
a ∈ E(G) there is no subgraph in HG,a isomorphic to G.
Proof. We use induction on the order of the graph G. If G ∼
= K2 then HG,a
is a pair of isolated vertices. Since the resulting graph is edgeless, the theorem
15
w*
v*
F
v'
w'
Figure 2.3: Cut vertices v 0 , v ∗ and non-cut vertices w0 , w∗ in HG,a − F
is true in the case of n(G) = 2. Now assume that the claim is true for every
connected graph with order strictly less than k. That is, if F is a connected
graph with order strictly less than k then there is no edge f ∈ E(F ) such that
F ⊆ HF,f .
Let G be a connected graph of order k with at least one edge a = xy such
that G ⊆ HG,a . Note that e(G − {a}) = e(G) − 1, and e(HG,a ) = 2e(G) − 2.
Hence it follows that there must be an edge α ∈ E(G − a) such that both α0
and α∗ are in this embedding of G. If no such edge exists then every edge in
e(G) − {a} appears at most once in the embedding and e(G) ≤ e(G) − 1, which
is impossible. We now show that there is a vertex w ∈ G − {x, y} such that
neither w0 now w∗ is in the embedding of G.
/ {x, y}. Since both α0 and α∗ are in
Case 1: Say that α = v1 v2 with v1 , v2 ∈
this embedding of G, it follows that v10 , v20 , v1∗ , v2∗ are as well. The embedding of
G contains n(G) − 4 other vertices, so at most n(G) − 2 remaining vertices of G
appear at least once in this embedding. So, there are at least two vertices in G
w, z ∈ V (G) such that none of {w0 , z 0 , w∗ , z ∗ } are in this embedding of G. Since
16
G is connected and {x0 , y 0 } = {x∗ , y ∗ } is a cut set of HG,a , at least one of {x0 , y 0 }
must be in this embedding of G. So, at least one of {w, z}, say w, is neither x
nor y. So there is a vertex w ∈ V (G) − {x, y} such that neither w0 nor w∗ is in
the embedding of G.
Case 2: Assume, without loss of generality, that v1 x. Since both α0 and α∗
are in the embedding of G, the embedding of G includes both x0 and x∗ = y 0 .
Since both v20 and v2∗ are in the embedding of G then by an argument analogous
to the one in the previous case there is a vertex w ∈ V (G) such that neither w0
nor w∗ are in the embedding of G. The vertex w is neither x nor y since both
x and y are in the embedding of G.
Note that the edge α cannot be incident to both x and y in G − {a} since
xy = a is a single edge in G.
So, in both cases there exists a vertex w ∈ V (G) − {x, y} such that neither
w0 nor w∗ is in the embedding of G. By Lemma (2.3.1) we can assume that w is
not a cut vertex of G. So, G − w is connected and contains the edge a, and the
embedding of G is also contained in HG−w,a . The graph G − w ⊆ G ⊆ HG−w,a
and n(G − w) < k, which contradicts our inductive assumption. Therefore, for
any connected graph G and single edge a ∈ E(G) there is no subgraph of HG,a
isomorphic to G.
Let D be an oriented graph. We want to show that there is some integer
ND such that for every n ≥ ND there is an oriented graph F on n vertices that
is D-saturated.
Lemma 2.3.3. If there is an oriented graph H on n vertices with non-adjacent
vertices x and y without a subgraph isomorphic to D, but D is a subgraph of both
17
H + xy and H + yx, then every tournament containing H also contains D. So,
there is an oriented supergraph of H on the vertices of H that is D-saturated.
Theorem 2.3.4. For every oriented graph D there is some integer ND such that
for every n ≥ ND there is a graph H on n vertices with non-adjacent vertices
x, y ∈ V (H) not containing a subdigraph isomorphic to D but with the property
that both H + xy and H + yx contain subgraphs isomorphic to D.
Proof. First consider the case where u(D) is connected. Let ND = 2n(D) − 2,
say a ∈ A(D), and let HD,a be the surgery as defined above. For n ≥ 2n(D) − 2,
define H to be the graph HD,a ∪ K̄n−ND , (see Figure 2.4). By Theorem 2.3.2,
u(D) is not a subgraph of u(HD,a ), and hence D is not a subdigraph of H.
However, the addition of either the arc xy or the arc yx creates a copy of D. By
Lemma 2.3.3, there is a D-saturated oriented graph on n vertices.
If u(D) is not connected then let J be the component of D with greatest
order. Let a be an arc in J and let H = HJ,a ∪ (D − J). Since HJ,a does not
contain a copy of J, H does not contain a copy of D.
Therefore, given any oriented graph D and integer n ≥ 2n(D)−2 sat(n, D), ex(n, D)
are well-defined.
2.4 Results
For each m define P~m to be the directed path on m vertices.
Theorem 2.4.1. sat(n, P~3 ) = n − 1
Proof. The lower bound is achieved by orienting all edges of K1,n−1 into the
center vertex, (see Figure 2.5). If any oriented graph D contains fewer arcs,
18
Figure 2.4: The union of HG,a with isolated vertices
Figure 2.5: The smallest P~3 -saturated oriented graph on 7 vertices
then it is not weakly connected and must contain at least 2 oriented trees T1
and T2 . Every oriented tree contains both a source and a sink, which can be
easily seen by considering the endpoints of any maximal directed path. If neither
tree contains a P~3 then neither does the graph with an added arc from a source
of T1 to a sink of T2 .
Theorem 2.4.2.
1.
sat(n, P~4 ) =


n
if 3|n

 n − 1 otherwise
19
Figure 2.6: A P~5 -saturated oriented graph
2.
3.



n
if 3|n



sat(n, P~5 ) ≤ n + 2 if n = 1 mod 3




 n + 4 if n = 2 mod 3



n
if 3|n



sat(n, P~6 ) ≤ n + 2 if n = 1




 n + 5 if n = 2
mod 3
mod 3
Proof. To see the upper bounds, consider the oriented graph composed of disjoint directed 3-cycles and possibly the following, depending on parity:
1. An isolated vertex or P~2 for sat(n, P~4 )
2. One or two strongly oriented K4 s for sat(n, P~5 ), (see Figure 2.6)
3. A strongly oriented K4 or K5 for sat(n, P~6 ).
For the sharpness of sat(n, P~4 ) we consider two cases. If 3 - n and D is a
P~4 -free oriented graph on n vertices with fewer than n − 1 arcs, then D has at
least 2 components that are oriented trees. Each contains a source and a sink,
and any added arc from the source of one to the sink of another will not create
a P~4 . Consequently if 3|n and D has fewer than n arcs, then there is at least
20
one component T that is an oriented tree, and T must contain a source and a
sink. If all sinks are out neighbors of all sources, then the component is an inor out-star which itself is not P~4 saturated. If there is a source not adjacent to
a sink then this added arc does not result in a P~4 . So, we have sharpness.
We now determine the exact value for the saturation number of an oriented
star.
Theorem 2.4.3. Let K1,(a,b) be the directed star with in-degree b and out-degree
a. If a 6= b and k = max{a, b} then sat(n, K1,(a,b) ) = n(k − 1)
Proof. Refer to the directed star K1,(a,b) as K. Say D is an oriented graph on
n vertices that is K-saturated. For any pair u, v of non-adjacent vertices we
require that D ∪ uv and D ∪ vu both contain a copy of K. So, either d− (u) ≥ b
and d+ (u) = a − 1, or d+ (v) ≥ a and d− (v) = b − 1, and either d− (v) ≥ b and
d+ (v) = a − 1, or d+ (u) ≥ a and d− (u) = b − 1. If d− (u) ≥ b and d+ (u) ≥ a, or if
d+ (v) ≥ a and d− (v) ≥ b, then D contains K and so D is not K-saturated. The
only 2 of the 4 possible combinations that don’t result in D already containing
K are when either both u and v are centers of a K1,(a−1,b) or both are centers of
a K1,(a,b−1) . We refer to this property as u and v being of the same type Without
loss of generality we can assume that a > b. So, any vertex v is either adjacent
to or from every other vertex, d+ (v) = a − 1, or a ≤ d+ (v) < n − 1. Since
the sum of all out-degrees is equal to the sum of all in-degrees, D contains at
least 21 n[(a − 1) + (a − 1)] = n(a − 1) arcs. This bound is realized by a regular
oriented graph on n vertices with degrees (a − 1, a − 1), in which the former
represents the out-degree of each vertex and the latter represents the in-degree
21
of each vertex, which is D-saturated.
Note that Theorem 2.4.3 only holds for orientation in which the out-degree
of the center vertex differs from its in-degree. If they are the same then we get
a different result.
Theorem 2.4.4. sat(n, K1,(a,a) ) = (n − 1)(a + 1)
Proof. Consider a (a−1, a−1)-regular oriented graph D on n−1 vertices, joined
from a single vertex v, (see Figure 2.7). Every vertex other than v has degree
(a, a − 1), and any new arc must be from one of these to another. So, D is
K1,(a,a) -saturated. Now let F be a K1,(a,a) -saturated oriented graph. Notice, as
in Theorem 2.4.3, that every vertex in F with a non-neighbor is at the center
of a K1,(a,a−1) or a K1,(a−1,a) , with non-adjacent vertices of the same type. Note
that every vertex must either be at the center of a K1,(a−1,a−1) or share an arc
with every other vertex. If every vertex has a non-adjacency, then, in order to
preserve equality of in- and out-degree sums in F , we must have at least one
arc xy or yx such that x is at the center of a K1,(a−1,a) and y is at the center of
a K1,(a,a−1) . Therefore x shares an arc with every vertex of the same type as y
and y shares an arc with every vertex of the same type as x. The number of arcs
added in this manner is minimized when the x-type or y-type sets of vertices is
of order 1. However, this yields a graph of the same size as our construction,
and hence we’ve minimized F .
We conclude this section with another definition and a familiar construction.
22
Figure 2.7: A K1,(3,3) -saturated oriented graph
Definition 2.4.5. Let G, H be graphs and let kG (H) denote the number of
subgraphs of H isomorphic to G. H is strongly G-saturated if for any edge
∈ H c kG (H + ) > kG (H).
Note that any G-saturated graph is strongly G-saturated, but the converse
is not necessarily true.
Theorem 2.4.6. If T Tm is the transitive tournament on m vertices then
m−2
sat(n, T Tm ) =
+ (n − m + 2)(m − 2).
2
Proof. Erdös, Hajnal, and Moon [21] showed that sat(n, Km ) =
m−2
2
+ (n −
m + 2)(m − 2) by considering Km−2 ∨ K̄n−m+2 . Orient the (m − 2)-clique to
obtain T Tm−2 and each arc from the remaining vertices to this tournament, (see
Figure 2.8). The resulting oriented graph F is T Tm -saturated and provides an
upper bound on sat(n, T Tm ). Now let J be a T Tm -saturated oriented graph
with x, y ∈ J non-adjacent vertices. Since J + xy, J + yx both contain T Tm , the
23
TTm
Figure 2.8: A transitive tournament saturated oriented graph, consisting of a
transitive tournament joined from a set of isolated vertices
addition of xy to u(J) creates a new copy of Km . Therefore, u(J) is strongly
Km -saturated. In [2] it is shown that Km−2 ∨ K̄n−m+2 is the unique smallest
strongly Km -saturated graph on n vertices. Therefore u(J) is Km−2 ∨ K̄n−m+2
and J has m−2
+ (n − m + 2)(m − 2) arcs. Therefore,
2
m−2
sat(n, T Tm ) =
+ (n − m + 2)(m − 2).
2
2.5 Some Upper Bounds
We begin our examination of bounds on sat with a definition followed by
an examination of path orientations.
Definition 2.5.1. An oriented graph D is hamiltonian connected if for any
pair x, y of vertices of D there is a hamiltonian path from x to y and from y to
x.
24
Let k ≥ 7 be an integer. We will now construct a hamiltonian connected
tournament of order k. Define the digraph Dk with vertices v0 , v1 , . . . , vk−1 and
arcs vi vi+1 , vi vi+2 , and vi vi+3 for all 0 ≤ i ≤ (k − 1) with addition modulo k.
Lemma 2.5.2. The digraph Dk is hamiltonian connected.
Proof. We require that for every ordered pair of vertices in Dk there is a hamiltonian path from the first to the second. Without loss of generality we may
assume that the ordered pair is of the form (v0 , vi ) for some 1 ≤ i ≤ (k − 1). If
i = (k − 1) then the path v0 v1 . . . vk−1 is hamiltonian. If i < (k − 1) is odd then
consider the path v0 v2 v4 . . . vi+1 vi+2 . . . vk−1 v1 v3 . . . vi . If on the other hand i is
even then the path v0 v1 v3 . . . vi+1 vi+2 . . . vk−1 v2 v4 . . . vi will suffice. Therefore,
there is a hamiltonian path from v0 to every other vertex in Dk , and thus Dk is
hamiltonian connected.
Because Dk is a hamiltonian connected digraph of order k ≥ 7, so is every
tournament of order k that contains Dk as a subdigraph. Therefore Lemma
2.5.2 implies the existence of hamiltonian connected tournaments of all orders
at least 7.
Theorem 2.5.3. Let n > m ≥ 9 be integers. Then, n ≤ sat(n, P~m ) ≤
m−2
2
+
2(n − m + 2).
Proof. First, note that the argument in the proof of Thm. 2.4.1 applies here as
well, so that no P~m -saturated oriented graph contains a pair of oriented trees. In
fact, a single oriented tree is either P~3 -saturated, and therefore not P~m -saturated,
or contains both a source and sink. Joining the source to the sink does not create
a P~m . Therefore n ≤ sat(n, P~m ).
25
Tm-2
Figure 2.9: A P~m -saturated oriented graph
For the upper bound consider the following oriented graph H(m, n), seen
in Figure 2.9. Let Tm−2 be a hamiltonian connected tournament on m − 2
vertices whose existence is guaranteed by Lemma 2.5.2, with vertices labelled
v0 , v1 , . . . , vm−3 . Add n − m + 1 vertices labelled r0 , r1 , . . . rn−m and adjacent
from v0 . Finally, add a new vertex u adjacent to v0 , r0 , . . . rn−m . The graph
H(m, n) is P~m -saturated. This can be seen by first noticing that the longest
path in H(m, n) has order m − 1, then examining the graph H(m, n) + a for any
arc a ∈ H(m, n)c . Let i, j be integers. If a = ri rj then the path vm−3 . . . v0 ri rj
is a P~m . If a = vi rj then consider the path uv0 P0i vi rj where P0i is a hamiltonian
path from v0 to vi in Tm−2 . Alternately, if a = ri vj then consider ri vj Pj0 r0 , unless
i = 0 in which case the path ri vj Pj0 r1 is a P~m . If a = vi u then let P be the path
of order m − 3 obtained from P0i by removing the vertex v0 . Then P vi uv0 r0 is a
path of order m. Finally, if a = uvi then the path uvi Pi0 v0 r0 suffices. Therefore,
H(m, n) is P~m -saturated and n ≤ sat(n, P~m ) ≤ m−2
+ 2(n − m + 2).
2
Note that if m is 7 or 8 then sat(n, P~m ) is bounded above by 23 n+c where c is
a constant depending on n (mod m). This bound is achieved by a construction
similar to that in Thm. 2.4.2 but composed of disjoint strong tournaments of
26
order 4 along with zero, one, two, or three strong tournaments of order 5 and,
in the former case, at most one directed 3-cycle.
Theorem 2.5.4. Let o(Pm ) be an orientation of Pm , with m = 2k, k > 8 and
k
n
odd. Then, sat(n, o(Pm )) ≤ (n (mod k)) k+1
+
b
c
.
2
k
2
Proof. Consider the oriented graph F consisting of disjoint rotational tournaments of order k. Havet and Thomassé [36] showed that every tournament on
at least 8 vertices contains every orientation of a hamiltonian path. So, each
vertex of F is an end vertex of every orientation of every path of order at most
the order of the component in which the vertex is contained. Thus, the addition
of an arc between vertices of different components will yield every orientation
of a path of order m. A similar result can be determined for m odd.
2.6 The minimum order of a D-saturated oriented graph
We mentioned in Section 2.1 that for a simple graph G, sat(n, G) is welldefined for any n ≥ n(G), and in Section 2.2 that this is not necessarily the
case for oriented graphs. Consider the case when D is a directed cycle on three
vertices, denoted C~3 . There are, up to isomorphism, only three oriented graphs
of order three and size two, (see Figure 2.10). For each, there is an arc whose
addition creates no C~3 . Consequently, sat(3, C~3 ) is undefined. However, the
directed cycle on four vertices is C~3 -saturated.
This leads naturally to wonder for which n sat(n, D) is defined. Given a
oriented graph D we refer to the smallest n such that sat(n, D) is defined as
ND . Our surgery in section 2 of this paper gives 2n(D) − 2 as an upper bound
on ND .
27
Figure 2.10: No oriented graph on 3 vertices is C3 -saturated
vm
v1
Figure 2.11: The addition of either arc v1 vm or arc vm v1 generates a hamiltonian cycle
For the transitive tournament T Tn on n vertices, the construction used in
Theorem 2.4.6 demonstrates that NT Tn = n.
Every vertex in a strong tournament on k vertices is both the initial and
terminal vertex of directed paths of order k. Therefore for m ≥ 4 the oriented
graph on m vertices composed of an isolated vertex and a strong tournament on
m − 1 vertices is P~m -saturated. A pair of isolated vertices is P~2 -saturated, and
three vertices with a pair of arcs that share an initial vertex is a P~3 -saturated
graph. So, NP~m = m for all m ≥ 2.
Similarly, for m > 3 we can show that NC~m = m. Consider a directed path
v1 , v2 , . . . vm on m vertices. Add the arcs v1 vm−1 and vm v2 , (see Figure 2.11).
The resulting graph does not contain a directed m-cycle, but the addition of
either v1 vm or vm v1 creates one, so by Lemma (2.3.3) there is a C~m -saturated
28
oriented graph on m vertices.
2.7 Future directions
I am also interested in applications to games on graphs and oriented graphs.
Ferrara, Harris, and Jacobson [25] determined the winner of F-saturator, an
impartial combinatorial game involving the avoidance of certain simple graphs.
I would like to investigate similar games involving avoidance of certain oriented
graphs.
29
3. Maximal Non-Interval Graphs and Bigraphs
This chapter represents joint work with B. Tonnsen.
3.1 Introduction
Definition 3.1.1. Let S be a family of open intervals on the real line, and G
a graph defined by the elements of S in the following way. For each interval
s ∈ S let vs be a vertex of G and join vertices vs , vt in G if and only if the
intervals s, t ∈ S have a non-empty intersection. We say that S is an interval
representation of the graph G. Every graph for which an interval representation
exists is called an interval graph.
Definition 3.1.2. A bipartite graph G = (X, Y ) is an interval bigraph if every
vertex can be associated with an open interval on the real line in which two
vertices x ∈ X, y ∈ Y are adjacent in G if and only if their associated intervals
have a non-empty intersection.
Note that we retain the bipartite property by not joining x1 , x2 ∈ X or
y1 , y2 ∈ Y even if their associated intervals intersect.
Interval graphs, and related families including interval bigraphs, proper interval graphs, and circular arc graphs, have been studied extensively, ([5], [6],
[7], [35], [38], [43], and [47]). We are primarily concerned with properties of these
families related to edge-maximality, which has thus far received little attention.
In [19] Eckhoff examines r-extremal interval graphs, those interval graphs with
the maximum number of edges among those with fixed order and clique number
30
r, and characterizes them. We extend the study of extremal interval graphs
to those graphs that are not interval graphs and have the maximum number
of edges, and additionally we examine related families. Although there is a
known, simple forbidden subgraph characterization for interval graphs, and an
as yet incomplete and very different forbidden subgraph characterization for interval bigraphs, we show that the family of extremal noninterval graphs and a
sub-family of extremal noninterval bigraphs are very similar.
We will use P to represent both a graph property and the complete family
of all graphs with property P.
Definition 3.1.3. Let P be a graph property and let G = (X, Y ) be a bipartite
graph not in P such that any edge = xy ∈ Gc with x ∈ X, y ∈ Y the graph
G + is in P. We say that G is P-bisaturated.
In section 2 we characterize all Pi -saturated graphs for Pi the family of
interval graphs. We then characterize all Pb -bisaturated graphs, up to a conjectured forbidden subgraph characterization, where Pb is the family of interval
bigraphs in section 3. In section 4 we address the family of edge maximal split
non-interval graphs. We end by examining unit interval and circular arc graphs.
3.2 Edge-Maximal Non-Interval Graphs
Let G be a graph. An asteroidal triple in G is a set A of three vertices such
that between any two vertices in A there is a path within G from one to the
other that avoids all neighbors of the third. An example is in the 3-sun in Fig.
3.1. Lekkerkerker and Boland showed that all interval graphs are completely
characterized by the absence of both asteroidal triples and induced cycles of
length greater than 3 in [43].
31
Figure 3.1: The 3-sun in which the white vertices form an AT
Lemma 3.2.1. Any graph containing an asteroidal triple contains an induced
P4 .
Proof. Let A = {x, y, z} ⊆ G be an asteroidal triple and assume that G does
not contain an induced P4 . Between any two vertices in A, say x, y there is
a shortest path P (x, y) in G avoiding the neighbors of the z. P (x, y) cannot
have length 1 since there is a z, y path that avoids all neighbors of x and thus
y cannot be a neighbor of x. If P (x, y) has length 3 or greater then it contains
an induced P4 . Therefore P (x, y), P (x, z), P (y, z) all have length 2. These three
paths are internally disjoint so G contains a 6-cycle xaybzca. If {a, b, c} are
not mutually adjacent then G contains an induced P4 . Thus G contains the
3-sun as a subgraph. If {a, b, c, x, y, z} does not induce the 3-sun then one of
the asteroidal paths P (u, v) between two vertices in A contains a neighbor of
the third. However, the 3-sun, and thus G, contains the induced P4 yacz.
Let Pi denote the set of interval graphs.
Theorem 3.2.2. Sat(n, Pi ) = {C4 ∨ Kn−4 }, n ≥ 4
32
Proof. Say G ∈ Sat(n, Pi ). By Lemma 3.2.1 G cannot contain an asteroidal
triple since the endpoints of an induced P4 can be joined to create an induced
C4 , which is another of our forbidden subgraphs. Therefore, G must contain an
induced cycle of length at least 4. If G contains an induced Ck , k > 4 then there
is a pair of vertices that when joined create an induced C4 . So, G must contain
an induced 4-cycle C. If there is an edge ∈ Gc − C c then G + still contains
the induced 4-cycle, and thus is not an interval graph. Therefore, G must be a
4-cycle joined to a complete graph.
Note that the family of maximally non-interval graphs is the collection of
cliques with an isolated pair of edges removed, precisely the same as the family of
maximally non-chordal graphs. While there are non-interval chordal graphs containing asteroidal triples, we have shown that no such graph is edge-maximally
non-interval.
3.3 Edge-Maximal Non-Interval Bigraphs
Throughout this section we are concerned with bipartite graphs G = (X, Y ).
We sometimes choose to discuss part X or Y , but since there is no distinction between parts X and Y this choice is merely made for convenience, and generality
should be assumed.
We begin by introducing some new structures.
Definition 3.3.1. [47] A set A = {a, c, e} of three edges of a graph G form an
asteriodal triple of edges (ATE) if for any two, say a, c, there is a path from one
endpoint of a to an endpoint of c that avoids all neighbors of the endpoints of e,
(see Fig. 3.2).
33
Figure 3.2: The bold edges form an asteriodal triple of edges
Figure 3.3: Three graphs, called insects, that contain an exobiclique
Definition 3.3.2. [38] Two sets A, B are incomparable if A * B and B *
A. An exobiclique is a bipartite graph H = (X, Y ) containing a biclique with
nonempty parts M ⊂ X and N ⊂ Y such that each of X −M and Y −N contain
three vertices with incomparable neighborhoods in the biclique, (see Fig. 3.3).
Definition 3.3.3. [38] An edge-asteroid of order 2k + 1 is a set of edges
e0 , e1 , ..., e2k such that, for each i = 0, 1, . . . , 2k, there is a path containing both
ei and ei+1 that avoids the neighbors of ei+k+1 ; the subscript addition is modulo
2k + 1, (see Fig. 3.4).
Note that the cycle C6 contains an asteroidal triple of edges but is not
an edge-asteroid of order 3, since for any 3 pairwise non-incident edges any
pair have an endpoint in the neighborhood of the third, (see Fig. 3.5). Thus,
34
e0
e1
e2
e3
e4
Figure 3.4: An edge asteroid of order 5. Note that joining either vertex in e4
to any vertex on the path between the two white vertices, inclusive, eliminates
this property.
Figure 3.5: An ATE that is not an EA of order 3
although their definitions are similar, an edge-asteroid is not simply a generalized
asteroidal triple.
None of the structures in Definitions 3.3.1, 3.3.2, 3.3.3 are permitted as
induced subgraphs in an interval bigraph, as seen by the following theorems of
Muller, Hell and Huang, and Harary et al.
Theorem 3.3.4. A bipartite graph containing an induced asteroidal triple of
edges or an induced cycle with length greater than 4 is not an interval bigraph.
[35]
Theorem 3.3.5. A bipartite graph containing an induced exobiclique or an edge
asteroid is not an interval bigraph. [38]
Let Pb be the family of interval bigraphs.
35
There is currently no forbidden subgraph characterization of interval bigraphs. However, the preceding theorems provide the most extensive known
collection of forbidden subgraphs. Though there are currently proposed bipartite graphs [37] that are not interval but do not fall into one of the cases addressed
in Theorems 3.3.4 and 3.3.5 the authors are not aware of any examples without
induced paths of length at least 5. Since joining the endpoints of an induced P5
creates an induced C6 no graph containing an induced P5 is Pb -saturated.
We will return to this issue at the end of this section.
Definition 3.3.6. Let G = (A, B), H = (C, D) be bipartite graphs.
The
bipartite joins of G and H are the graphs consisting of a copy of G, a copy
of H, and either all adjacencies between A and C and between B and D, or all
adjacencies between A and D and between B and C. Denote by G ∗ H the family
of bipartite joins of G and H.
Lemma 3.3.7. Let G = (X, Y ) be a Pb -bisaturated bipartite graph of order n.
If G contains an induced C6 then there is an integer m < n such that G is a
graph in C6 ∗ Km,n−m−6 .
Proof. Assume not, and let C be an induced C6 in G. There is a non-adjacent
pair of vertices x ∈ X, y ∈ Y such that without loss of generality x ∈
/ C. The
addition of edge xy to G creates an induced C6 and hence G is not an interval
bigraph. So, G must be of the form described.
Lemma 3.3.8. No Pb -bisaturated bipartite graph contains an induced P6 .
Proof. Let G be a bipartite graph with v1 , v2 , . . . , v6 an induced P6 . The addition
36
of edge v1 v6 creates an induced C6 in the new graph, and hence G is not Pb bisaturated.
Lemma 3.3.9. Every asteroidal triple of edges contains either an induced P6
or an induced C6 .
Proof. Let G = X ∪ Y be a bigraph with an asteroidal triple of edges, a = a1 a2 ,
b = b1 b2 , and c = c1 c2 such that a1 , b1 , c1 ∈ X and a2 , b2 , c2 ∈ Y . Assume
there is no induced P6 or C6 in G, so a, b, and c cannot be alternating edges of
a C6 . Therefore, we assume there exists at least one minimal asteroidal triple
path, P 1 , with length at least 2. Without loss of generality assume this path is
between edges a and b. If P 1 has 3 or more edges then it comprises an induced
P6 with edges a and b, so assume this path between a and b has length 2, and
label the vertices a1 , d, b1 . There exists a minimal path from edge b to c that
avoids the neighborhood of a, call it P 2 . Again if P 2 has 3 or more edges then G
contains an induced P6 , so p2 has 1 or 2 edges. If P 2 has length 1 and the path is
b1 c2 , then the edge c1 a2 either exists, creating an induced C6 with a2 a1 db1 c1 c1 a2 ,
or does not, in which case the vertices form an induced P6 . Now assume the
edge comprising P 2 is b2 c1 . In this situation, there are no more edges among
these 6, and a2 a1 db1 b2 c1 is an induced P6 . Therefore, let us assume that P 2
has length 2. If P 2 = b1 ec1 then, since there are no more edges among these
6 vertices, a1 db1 ec1 c2 is an induced P6 . So assume the path is b2 ec2 . The only
edges that would not eliminate the property that G contains an asteroidal triple
of edges are ed and a2 c1 . If ed, a2 c1 ∈ E(G), then c1 a2 a1 dec2 c1 is an induced C6 .
Otherwise, if neither edge is in G then a2 a1 db1 b2 e is an induced P6 . Finally, if
ed ∈ E(G), but a2 c1 ∈
/ E(G), then a2 a1 dec2 c1 is an induced P6 . If a2 c1 ∈ E(G),
37
but ed ∈
/ E(G), then the 6 vertices create an induced C8 , which contains an
induced P6 . Therefore, G has an induced P6 or C6 .
By Lemmas 3.3.8 and 3.3.9 no bipartite graph with an asteroidal triple of
edges is Pb -bisaturated unless it is of the form C6 ∗ Kl,m for integers l, m.
Lemma 3.3.10. Every exobiclique contains either an induced P6 or an induced
C6 .
Proof. Let G = (A, B) be an exobiclique with the property that for any edge
∈ Gc , G + is not an exobiclique. There are sets X, N ⊆ A and Y, M ⊆ B
such that H = (M, N ) is a biclique and X, Y are sets of size at least 3 with
incomparable neighborhoods. There exist x1 , x2 ∈ X with neighbors m1 , m2 ∈
M , respectively, such that x1 m2 and x2 m1 . Similarly there exist y1 , y2 ∈ Y
with neighbors n1 , n2 ∈ N , respectively, such that y1 n2 and y2 n1 . If x1 y2
or x2 y2 is an edge in G, then m1 n1 m2 x2 y2 x1 m1 is an induced C6 . Otherwise,
one of these edges can be added without eliminating the property that G is an
exobiclique, and thus G remains a non-interval bigraph.
By Lemmas 3.3.8 and 3.3.10 no bipartite graph containing an exobiclique is
Pb -bisaturated unless it is of the form C6 ∗ Kl,m for integers l, m.
Now consider G = (X, Y ), a bipartite graph containing an edge asteroid
{e0 , e1 , . . . , e2k } of order 2k + 1. For each i ei is the edge joining xi ∈ X, yi ∈ Y .
If the distance between the sets {xi , yi } and {xj , yj } is greater than 2 for some
0 ≤ i, j ≤ 2k then G contains an induced P6 , so we need only consider the case
in which the distance between any pair of edges in an edge asteroid is at most
38
2. Note that for all 0 ≤ i ≤ 2k we have that no endpoints of either ei or ei+1 is
a neighbor of either endpoint of ei+k+1 .
Lemma 3.3.11. If k = 1 then G contains an induced P6 .
Proof. Let {x0 y0 , x1 y1 , x2 y2 } be an edge asteroid of order 3 in G and assume
that G contains no induced P6 . The endpoints xi and yj are not adjacent for
any distinct pair i, j ∈ {0, 1, 2}, so without loss of generality there is a path
y0 ay1 in G for some vertex a that is not the endpoint of an edge in the EA.
Similarly there is a distinct vertex b that is not an endpoint of an edge in the
EA such that either y1 by2 or x1 bx2 is a path in G. Therefore, x0 y0 ay1 by2 or
x0 y0 ay1 x1 b is an induced P6 in G.
Now assume that k > 1. First, consider the case in which there is some i
such that yi ∼ xi+k+2 .
Lemma 3.3.12. If yi+k+1 ∼ xi+k+2 and xi ∼ yi+1 then G is not Pb -bisaturated.
Proof. Let 1 = yi xi+k+2 , 2 = yi+1 xi+k+1 , 3 = yi+k+1 xi+k+2 . Note that 1 , 3 are
edges in G and 2 is not, (Fig. 3.6). The addition of 2 , however, results in the
asteroidal triple of edges {1 , 2 , 3 }. By Lemma 3.3.9 G is not Pb -bisaturated.
Lemma 3.3.13. If yi+k+1 xi+k+2 or xi yi+1 then G is not Pb -bisaturated.
Proof. In the former case let = xi+1 yi+k+1 , and in the latter let = yi+1 xi+k+1 .
Let 1 = yi xi+k+1 , 2 = ei+1 , 3 = ei+k+1 . The addition of edge results in the
asteroidal triple of edges {1 , 2 , 3 }. By Lemma 3.3.9, G is not Pb -bisaturated.
39
xi
yi
x i+1
yi+1
x i+k+1
xi+k+2
yi+k+1 yi+k+2
Figure 3.6: The addition of edge {xi+1 , yi+k+1 } creates an ATE
Now, consider the alternate case in which there is no i such that yi ∼ xi+k+2 .
Lemma 3.3.14. Let G = (X, Y ) contain an edge asteroid {e0 , e1 , . . . , e2k } of
order greater than three with the property that for all i, yi xi+k+2 and xi yi+k+2 . Then, G is not Pb -bisaturated.
Proof.
Case 1: Assume that there is an i such that a shortest path between ei and ei+k+1
includes the vertex ai , a shortest path between ei+1 and ei+k+2 includes
the vertex ai+1 6= ai , ei is not adjacent to ai+1 , and ei+1 is not adjacent to
ai . Then, we have one of the graphs in Fig. 3.7 as a subgraph of G. There
is a path from the endpoints of ei to the endpoints of ei+1 that avoids the
neighbors of the endpoints of ei+k+1 , and thus both graphs in the figure
contain an induced P6 that includes as a subpath yi ai yi+k+1 xi+k+1 . By
Lemma 3.3.8 G is not Pb -bisaturated.
Case 2: Now say that for all i there is a vertex ai such that ei , ei+1 , ei+k+1 ∼ ai , as
depicted in Figure 3.8.
40
xi
yi
x i+1
yi+1
ai
x i+k+1
xi+k+2
xi
yi+k+1 yi+k+2
yi
x i+1
yi+1
ai+1
x i+k+1 xi+k+2
yi+k+1 yi+k+2
ai
ai+1
Figure 3.7: Possible configurations of the vertices ai , ai+1
xi
yi
x i+1
yi+1
ai
x i+k+1
xi+k+2
yi+k+1 yi+k+2
ai+1
Figure 3.8: ai , ai+1 in the same partite set
Case 2a: There is an integer i such that ai and ai+1 are distinct vertices, (Fig.
3.9). If ai , ai+1 are in the same partite set of G, set X, then let
= xi yi+k+2 . The addition of to G results in the asteroidal triple
of edges {ei , ei+1 , ei+k+2 }.
If ai ∈ X, ai+1 ∈ Y , (Fig. 3.9), then either ai ai+1 , in which
case ai yi+1 xi+1 ai+1 xi+k+2 yi+k+2 forms an induced P6 , or ai ∼ ai+1 , in
which case the addition of the edge xi yi+k+2 completes an asteroidal
triple of edges among {ei , ei+k+2 , ai ai+1 .
Case 2b: There is a vertex a in G that satisfies ai = a for all 0 ≤ i ≤ 2k. Let A
41
ai+1
xi
x i+1 x i+k+1
xi
xi+k+2
x i+1
2
1
3
yi+1
x i+k+1 xi+k+2
2
1
yi
ai
yi+k+1 yi+k+2
yi
ai
3
yi+1
yi+k+1 yi+k+2
ai+1
Figure 3.9: Possible configurations when ai , ai+1 in different partite sets
be the collection of all such vertices, i.e. A = {a ∈ V (G) : a ∼ ei ∀i}.
G0 = G − A has the same edge asteroid as G since no edge asteroid
path includes a vertex in A, and G0 contains no vertex adjacent to all
edges of the edge asteroid. So, G0 falls into Case 1 or Case 2a above,
and is an induced subgraph of G.
Lemmas 3.3.7 through 3.3.14 lead us to the following conjecture.
Conjecture 3.3.15. G is Pb -bisaturated if and only if G ∈ C6 ∗ Kl,m for some
integers l, m.
Note that this family of maximally non-interval bigraphs is precisely the
family of bicliques with 3 isolated edges removed.
Thus far we have assumed the forbidden subgraph characterization implied
by Theorems 3.3.4 and 3.3.5. While there are current attempts at constructing
examples of graphs that potentially invalidate this characterization, they are all
quite large. Because complete bipartite graphs are interval bigraphs any noninterval bigraph that avoids the aforementioned forbidden subgraphs contains a
42
a
b
G
G
1
3
Figure 3.10: The forbidden subgraphs G1 and G3
high number of non-adjacencies. Therefore, we expect that any bipartite graphs
that violate the characterization will contain rather long induced paths. We
have examined one such graph, briefly mentioned at the top of page 323 in [38],
and supplied to us [37] by the authors of [38], and we have found it to contain
an induced P7 .
3.4 Edge-Maximal Split Non-Interval Graphs
Definition 3.4.1. A graph G is a split graph if the vertices can be partitioned
into sets A and B such that the induced subgraph on A is a complete graph and
the the induced subgraph on B an independent set.
See Fig. 3.10 for examples in which the white vertices represent independent
sets and the black vertices cliques.
Let Ps denote the property of being an interval graph or a non-split graph.
Let G1 and G3 be the graphs in Fig. 3.10, and let G2 denote the 3-sun in Fig.
3.1.
Foldes and Hammer showed [27] that a split graph is interval if and only
if it does not contain an induced subgraph isomorphic to one of the graphs
G1 , G2 , G3 .
43
Lemma 3.4.2. G1 , G2 ∈ Sat(n, Ps ), but G3 is not.
Proof. G1 and G2 are both split graphs that contain asteroidal triples denoted
by {x, y, z} in Figure 3.1 and the white vertices in Figure 3.10. Any edge that
can be added to either graph without resulting in an interval graph is an edge
that eliminates the split property. However, the addition of edge ab to G3 ,
(see Fig. 3.10), does not destroy the asteroidal triple nor result in a non-split
graph.
Theorem 3.4.3. Sat(n, Ps ) = {G1 ∨ Kn−6 , G2 ∨ Kn−6 }, n ≥ 6
Proof. Say G ∈ Sat(n, Ps ). One of {G1 , G2 } must therefore be a subgraph of G.
If n > 6 then G also contains least one other vertex v. The smaller graph G − v
is a split graph, with vertices appropriately partitioned into sets with induced
subgraphs A, a clique, and B, an independent set. Let u be a neighbor of v. If
u is in B then in order for G to be a split graph A ∪ {v} must be a complete
graph. We can also join v to every other vertex in B without destroying either
condition. So, v is is adjacent to every vertex in G − {v}. If u is in A then either
< V (A)∪{v} > is a clique, which again implies that v is adjacent to every vertex
in G − {v}, or there is a vertex a ∈ A such that v a and V (B) ∪ {v} is an
independent set. In this case the edge va can be added to G without violating
the split property, and therefore A ∪ {v} must be a clique. Hence v is adjacent
to every vertex in B in order for G to be P-saturated.
Therefore, Sat(n, Ps ) = {G1 ∨ Kn−6 , G2 ∨ Kn−6 } when n ≥ 6
3.5 Other Examples
44
Definition 3.5.1. A graph G is a unit interval graph if it has an interval representation in which every vertex is associated with an interval of length 1.
The following theorem of Roberts [53] will be useful.
Theorem 3.5.2. An interval graph G is a unit interval graph if and only if it
does not contain K1,3 as an induced subgraph.
Let Pu denote the family of unit interval graphs.
Theorem 3.5.3. Sat(n, Pu ) = {K1,3 ∨ Kn−4 , C4 ∨ Kn−4 }, n ≥ 4
Proof. Let n ≥ 4. We have already seen in Theorem 3.2.2 that C4 ∨ Kn−4 is
an edge maximal non-interval graph. Since the addition of an edge results in a
clique with a single edge removed, which does not contain an induced K1,3 , it is
also in Sat(n, Pu ). K1,3 ∨ Kn−4 is an interval graph, but by Theorem 3.5.2 is not
unit interval. However, any edge added to K1,3 ∨ Kn−4 , (in fact, there is only
one such edge without loss of generality), eliminates the induced K1,3 without
creating either an asteroidal triple or a large induced cycle. So, this graph is
also in Sat(n, Pu ).
Now say that G is a Pu -saturated graph. Then G must contain either a
large cycle, an asteroidal triple, or a K1,3 as an induced subgraph. Denote this
induced subgraph J. By Lemma 3.2.1 we know G cannot contain an asteroidal
triple, and by Theorem 3.2.2 there is no induced cycle with length greater than
4. So, G contains either an induced K1,3 or an induced C4 . If G is not precisely
K1,3 ∨ Kn−4 or C4 ∨ Kn−4 , then it is a proper subgraph of one of them. There is
an edge that can be added to G without eliminating J as an induced subgraph.
Hence, Sat(n, Pu ) = {K1,3 ∨ Kn−4 , C4 ∨ Kn−4 }
45
a
a
a
c
b
c
b c
bipartite claw
c
a
b
umbrella*
...
n-net
net*
c
e
d
1 2
..... n
1 2
(n-tent)*
b
n
c
1
n
2
n-1
n-2
C*
n
Figure 3.11: The basic minimally non-circular arc graphs
We now introduce a type of graph often studied alongside interval graphs.
Definition 3.5.4. A graph G is called a circular arc graph if it is an interval
graph of a family of arcs of a circle.
In other words, instead of modeling intervals in the real line, the vertices of
a circular arc graph can be represented by arcs on a circle. Note that all interval
graphs are easily seen to be circular arc graphs by applying an isomorphism
from the real line to the unit circle minus a point.
A graph G is minimally P if it has property P but no proper subgraph has
property P. Denote by G∗ the graph obtained from G by adding an isolated
vertex. We will use the following Lemma from [55].
Lemma 3.5.5. The following graphs are minimally non-circular arc graphs:
bipartite claw, net∗ , n-net for all n ≥ 3, umbrella∗ , (n-tent)∗ for all n ≥ 3, and
Cn∗ for every n ≥ 4. Any other minimally non-circular arc graph is connected.
46
Figure 3.12: A graph and its circular arc representation
Bonomo et al. refers to the graphs above (see Fig. 3.11) as basic minimally
non-circular arc graphs in [5]. Let Pc denote the family of circular arc graphs.
Theorem 3.5.6. The only Pc -saturated basic minimally non-circular arc graph
is C4∗ .
Proof. It is easy to show that C4∗ is Pc -saturateed, since the addition of any edge
results in either a C4 with a pendant edge, realizable by the arcs
(0, 2.1), (2, 4.1), (4, 6.1), (6, 0.1), (1, 1.5)
in radians (Fig. 3.12), or (K4 − e) ∪ K1 , which is an interval graph and therefore
a circular arc graph. Now let G be a bipartite claw, net∗ , n-net or umbrella∗ .
The addition of the edge ab as labeled in figure 3.11 creates an induced C4∗ from
the ab-path, edge ab, and vertex c. In the n-tent if we add the edge joining
vertices 1 and n, then we get an induced C4∗ from the vertices: 1, d, e, n, and c.
If n ≥ 5 in the Cn∗ , then the addition of the edge between vertices 1 and n − 2
creates an induced C4∗ from the vertices: 1, n, n − 1, n − 2, and c. Therefore, the
only basic minimally non-circular arc graph in Sat(n, Pc ) is C4∗ .
47
4. Subdivided Graph Saturation
This chapter represents joint work with M. Ferrara, M. Jacobson, K. Milans,
and P. Wenger.
4.1 Introduction
We begin this chapter with some terminology.
Definition 4.1.1. A maximal 2-connected subgraph B of a graph G is a block
of G. If B ∼
= H for some H, then we will say that B is an H-block of G.
Definition 4.1.2. If X a cut-set of G, then an X-lobe of G is any subgraph
induced by X and the vertices in a single component of G − X. An H-lobe of
G is a lobe of G that is isomorphic to H.
We let G− denote the graph obtained by deleting all of the pendant vertices
of G.
Recall from Chapter 1 that a graph G can be subdivided by the replacement
of edge {x, y} with vertex z and edges {x, z}, {z, y}. We let S(G) represent the
family of all graphs resulting from any number of subdivisions of G, including
G itself.
Definition 4.1.3. Let G be a graph, F ∈ S(G). We can view V (G) as a subset
of V (F ). We call these vertices the branch vertices of F .
The focus of this chapter is to study S(H)-saturated graphs for certain
choices of H with the goal of analyzing the similarities and differences between
48
H-saturated and S(H)-saturated graphs of minimum size. In particular, we
consider the cases where H is either a clique or a cycle and we obtain a number
of exact results and bounds.
4.2 Cycles
4.2.1 History
Ct -saturated graphs of minimum size have been studied extensively. While
a number of bounds have been determined, very few precise values of sat(n, Ct )
are known. Some known results are summarized in the Table 4.1.
Recall that a graph of order n is hamiltonian if it contains a cycle of length
n.
Definition 4.2.1. A graph G of order n is maximally non-hamiltonian (MNH)
if it is not hamiltonian but the addition of edge {x, y} for any non-adjacent pair
of vertices x, y results in a hamiltonian cycle.
The Coxeter graph [14] is a MNH graph of order 28 and smallest cycle length
7, (Fig. 4.1).
Using an edge coloring argument and Isaacs’ flower snarks from [39], Clark
Crane, Entringer, and Shapiro determined the exact value of sat(n, Cn ) for sufficiently large n [12], [13], [11]. It is their method that we enlist in this section.
4.2.2 t ≤ 6
We begin by determining sat(n, S(Ct )) exactly for t ≤ 6.
We begin by noting that any cycle is a subdivision of C3 , and therefore
sat(n, S(C3 )) = n − 1 for all n. Next we consider S(C4 )-saturated graphs.
49
t sat(n, Ct )
3 n − 1 [21]
c, n ≥ 5 [48, 57]
4 b 3n−5
2
5 = d 10
(n − 1)e, n ≥ 11, n 6∈ N0 [9]
7
(n − 1)e − 1, n ∈ N0 = {11, 12, 13, 14, 16, 18, 20}
= d 10
7
l
l
≤ n + c nl for a constant c, l 6= 8, 10, n sufficiently large
≥ (1 +
1
)n, n
2l+8
≤ (1 +
2
)n
l−2
+
5l2
,l
4
≡ 0(mod2), l ≥ 10, n ≥ 31
≤ (1 +
2
)n
l−3
+
5l2
,l
4
≡ 1(mod2), l ≥ 17, n ≥ 71 [30]
6 ≤
3n
2
7 ≤
7n+12
5
l = 8, 9, 11, 13, 15 ≤
3n
2
≥ l ≥ 5 [1]
[1]
+
[1]
l2
2
[30]
e, n ≥ 20 even or n ≥ 17 odd [4, 11, 12, 13, 44]
n d 3n
2
Table 4.1: Known bounds and and exact values for sat(n, Ct )
A (1, 3)-tree is any tree T in which all vertices of T have degree either one
or three. Let Tn be the family of graphs of order n ≥ 3 vertices generated by
simultaneously performing ∆ − Y exchanges on all of the vertices of degree three
in T , and then deleting up to three pendants. An example of a member of this
family can be seen in Figure 4.2. We note here that every T ∈ Tn has size
t(n) = n + b n−3
c.
4
The following Lemma is a consequence of the elementary fact that any 2connected graph of order at most four contains a cycle of length at least four,
50
Figure 4.1: The Coxeter graph
Figure 4.2: A (1, 3)-tree and the associated graph in T18 .
along with the observation that every block of a S(C4 )-saturated graph must
also be S(C4 )-saturated.
Lemma 4.2.2. If G is a S(C4 )-saturated graph, then every block of G is a clique
of order at most three.
We also require the following Lemma, which suffices to establish that any T
in Tn is S(C4 )-saturated.
51
Lemma 4.2.3. If G is a graph satisfying both of the following two properties
then G is S(C4 )-saturated.
1. Every block of G is isomorphic to either K2 or K3 , and
2. no two K2 -blocks of G share a vertex,
Proof. Let G be as given in the statement of the Lemma and let u and v be
nonadjacent vertices in G. Properties (1) and (2) imply that, regardless of the
choice of u and v, there is a u − v path of length at least three in T , implying
that T + uv has circumference at least four.
Theorem 4.2.4. For n ≥ 1, sat(n, S(C4 )) = t(n) = n + b n−3
c.
4
Proof. The upper bound follows from Lemma 4.2.3, as any T ∈ Tn is S(C4 )saturated, as are K1 and K2 .
To establish the lower bound, let G be an S(C4 )-saturated graph of minimum
size. The Theorem holds if n ≤ 3, since in this case G must be a clique so we
may assume that n ≥ 4. Lemma 4.2.2 implies that G must have more than
one block, so first suppose that xy and xz were distinct K2 -blocks of G. Then
each block of G + xy must still be either a K2 or a K3 , implying that G cannot
contain a cycle of length at least four, and hence that G is not S(C4 )-saturated.
Thus, no two K2 -blocks of G share a vertex.
Next, assume that J and K are K3 -blocks of G sharing a cut-point x, and
furthermore that for some vertex y, xy is an end-block of G. Then we can remove
y and expand x to a K2 without increasing the size of G. If x does not lie in a
K2 -block of G, but there is some other pendant vertex y in G, then we can again
52
delete y from G and expand x to a K2 without increasing the order or size of G.
If G does not contain any pendant vertices, then G contains distinct end-blocks
B and B 0 isomorphic to K3 . We can therefore expand x to a K2 , remove two
non-cut-vertices from B and add a pendant vertex to a non-cut-point vertex of
B 0 . Lemma 4.2.3 implies that, throughout each of these operations, G remains
S(C4 )-saturated. Therefore, we can assume that κ(G) = 1 and each cut-point
in G is contained in a K2 -block and a K3 -block of G.
If G contains two end-blocks isomorphic to K3 , then by removing the two
non-cut-point vertices of one and adding pendant vertices to each of the non-cutpoint vertices of the other results in an S(C4 )-saturated graph with the same
order as G and fewer edges, contradicting our minimality assumption. Thus,
we can also assume that G contains at most one end-block isomorphic to K3 .
Together with the previous assumptions, we conclude that G ∈ Tn , completing
the proof.
We now turn our attention to S(C5 )- and S(C6 )-saturated graphs. Our
technique for determining the saturation numbers of these families is similar to
that employed in Theorem 4.2.4. We call Bt ∼
= K2 ∨ K t , a book. We refer to
the edge uv that is joined to the maximum independent set as the spine of the
book, and to the t vertices in the independent set as the pages of the book.
Lemma 4.2.5. If G is a 2-connected S(C5 )-saturated graph of order n that is
not a clique, then n ≥ 5 and G ∼
= Bn−2 .
Proof. Suppose first that G is 3-connected. Since G is not a clique, then G must
have order at least five. If G has order strictly greater than five, then Menger’s
53
Theorem and a result of Chvátal and Erdős [12] imply that G must contain a
cycle of length at least six. Hence we conclude that G has order exactly five,
and since δ(G) ≥ 3, Dirac’s Theorem [16] implies that G is hamiltonian, again
a contradiction.
Hence we conclude that κ(G) = 2 and consider a minimum cut set X =
{x, y} in G. Each X-lobe of G must contain an x − y path of length at least
two. If any of these paths had length more than two, then G would contain a
cycle of length at least five. Since G contains no cut vertices, we conclude that
each X-lobe has order exactly three, and hence that G is isomorphic to either
Bn−2 or K2,n−2 . Since adding the edge xy in the latter case does not create a
cycle of length at least five, we conclude that G ∼
= K3
= Bn−2 , as desired. As B1 ∼
and B2 ∼
= K4 − e are not S(C5 )-saturated, we conclude that n ≥ 5.
Lemma 4.2.5 implies that every block of an S(C5 )-saturated graph is either a
complete graph of order at most four or a book with at least three pages. Next,
we consider the feasible placement of K2 -blocks within an S(C5 )-saturated graph.
Lemma 4.2.6. Let G be a S(C5 )-saturated graph and let B = xy be a block of
G isomorphic to K2 . Then if x is a cut-vertex of G, and B 0 6= B is a block of
G that contains x, either B 0 ∼
= K4 or B 0 ∼
= Bt and x does not lie on the spine
of B 0 . Consequently, G · xy is also S(C5 )-saturated.
Proof. Suppose that, without loss of generality, d(x) > 1 and that xz, z 6= y is
an edge in some block of G that is either K2 or K3 . Then in G + yz, the block
containing x, y and z would be either a K3 or a K4 − e. As the remaining blocks
from G would be unchanged, G + yz could not contain a cycle of length at least
54
five, contradicting the assumption that G was S(C5 )-saturated.
Suppose that xy is a block of G isomorphic to K2 and furthermore that x
lies in a block of G isomorphic to Bt . Then the edge xw cannot be the spine of
the book, as in G + wy, the block containing x, y and w would then be Bt+1 .
This again contradicts the assumption that G is S(C5 )-saturated.
Since each block of G must be S(C5 )-saturated, G · xy must be S(C5 )saturated if d(y) = 1. If d(y) > 1, then the fact that G · xy is S(C5 )-saturated
follows from several observations. First, the vxy -lobes of G · xy must be S(C5 )saturated. Secondly, since xy was only incident to blocks isomorphic to K4 or
Bt , t ≥ 3, one can readily verify that for any two vertices u and v lying in
distinct components of (G · xy) − vxy there is a u − v path of length at least four
in G · uv.
The following Lemma follows easily from Lemmas 4.2.5 and 4.2.6 and states
that, under the restrictions for K2 -blocks established in Lemma 4.2.6, we may
rearrange the blocks of an S(C5 )-saturated graph as needed.
Lemma 4.2.7 (The Rearranging Lemma). Let G denote the family of graphs H
with the following properties:
1. Every block of H is either a clique of order at most four or a book with at
least three pages, and
2. for any K2 block B of H and block B 0 6= B of H such that B ∩ B 0 6= ∅,
either B 0 is a K4 -block of H or B 0 is a Bt -block of H with t ≥ 3 such that
B ∩ B 0 is a page of B 0 .
Then a graph G is S(C5 )-saturated if and only if G ∈ G.
55
We now are ready to determine sat(S(C5 ), n).
Theorem 4.2.8. For n ≥ 7,
sat(n, S(C5 )) = f (n) =

10n 

− 1 n ≡ 0, 1, 3, 5 (mod 7)


 7




 10n
7
n ≡ 2, 4, 6 (mod 7)
Proof. We establish the upper bound by using the graphs H1 , . . . , H6 pictured
in Figure 4.3 to construct S(C5 )-saturated graphs Gn of order n for each n ≥ 7.
v
H0
H1
H2
H3
H4
H5
H6
Figure 4.3: S(C5 ) builders (subscripts indicate value of n (mod 7))
For 7 ≤ n ≤ 13, we let Gn = Hi+7 for each of the Gi in Figure 4.3. For
n ≥ 14 we construct Gn from Gn−7 by taking a copy of H1 and associating
the vertex v with the vertex in the copy of Hn−7
(mod 7)
contained in Gn−7 that
is marked with an open circle. It is straightforward to verify that each Gn is
S(C5 )-saturated for each n ≥ 7. We also note that these graphs are not unique.
56
As constructed, the block-cutpoint graph of each G−
n is a star with at most one
edge subdivided. It would have been acceptable to assemble the blocks of G−
n
so that the block-cutpoint graph was, for instance, a caterpillar with a longer
spine.
To establish the upper bound, we proceed by induction on n. It is straightforward to verify that, given Lemma 4.2.7, the graphs H0 , . . . , H6 serve to establish the base cases of 7 ≤ n ≤ 13. We therefore assume that for all 14 ≤ t < n,
sat(t, S(C5 )) = f (t) and consider G, a S(C5 )-saturated graph of order n ≥ 7 and
minimum size. We furthermore assume that e(G) < f (n). Note that G must
be connected, but by Lemma 4.2.5 cannot be 2-connected, as e(Bn−2 ) ≥ e(Gn )
for n ≥ 7. Thus G must have connectivity one, so that each block of G is either
a clique of order at most four or a book with at least three pages. Table 4.2,
which relates the values of f (n) and f (n − k) for 0 ≤ k ≤ 6, will be useful as we
proceed.
Suppose first that there is a cut vertex x in G such that some x-lobe L is
isomorphic to H1 . Then G0 = G − (L − x) is still S(C5 )-saturated by Lemma
4.2.6. Then G0 has at least f (n−7) edges by the induction hypothesis, and hence
G has at least f (n − 7) + 10 = f (n) edges, a contradiction to our assumption
about the minimality of G.
Similarly, now assume that there is some x-lobe L that is isomorphic to
H0 , and let p 6= x be a pendant vertex in L. Note that Lemma 4.2.6 implies
that G − p is S(C5 )-saturated, as must be G0 = G − (L − x). Therefore, the
induction hypothesis implies |E(G − p)| ≥ f (n − 1) and |E(G0 )| ≥ f (n − 6).
We now consider two possibilities, based on the value of n (mod 7). If n ≡ 1, 3
57
n (mod 7)
0
1
2
3
4
5
6
f (n − 1)
f (n) − 1 f (n) − 1 f (n) − 2 f (n) − 1 f (n) − 2 f (n) − 1 f (n) − 2
f (n − 2)
f (n) − 3 f (n) − 2 f (n) − 3 f (n) − 3 f (n) − 3 f (n) − 3 f (n) − 3
f (n − 3)
f (n) − 4 f (n) − 4 f (n) − 4 f (n) − 4 f (n) − 5 f (n) − 4 f (n) − 5
f (n − 4)
f (n) − 6 f (n) − 5 f (n) − 6 f (n) − 5 f (n) − 6 f (n) − 6 f (n) − 6
f (n − 5)
f (n) − 7 f (n) − 7 f (n) − 7 f (n) − 7 f (n) − 7 f (n) − 7 f (n) − 8
f (n − 6)
f (n) − 9 f (n) − 8 f (n) − 9 f (n) − 8 f (n) − 9 f (n) − 8 f (n) − 9
Table 4.2: f (n − k), 0 ≤ k ≤ 6, listed by value of n (mod 7).
or 5 (mod 7), then Table 4.2 gives that f (n − 1) = f (n) − 1. Thus, |E(G)| =
|E(G − p)| + 1 ≥ f (n − 1) + 1 = f (n), a contradiction. Similarly, if n ≡ 0, 2, 4
or 6 (mod 7), then f (n − 6) = f (n) − 9. Consequently, in this case, |E(G)| =
|E(G0 )| + 9 ≥ f (n − 6) + 9 = f (n), again a contradiction. Hence we may assume
that G has no H0 - or H1 -lobe.
We can also use Table 4.2 to show that G has no K3 - or K4 -lobes. Suppose
otherwise, so that there was some cut-vertex x in G such that the vertices
x, y and z formed a K3 -lobe of G. By the Rearranging Lemma, G − {y, z} is
still S(C5 )-saturated. However, by the induction hypothesis, |E(G − {y, z})| ≥
f (n − 2) and by Table 4.2, f (n − 2) ≥ f (n) − 3. Consequently, |E(G)| − 3 =
|E(G − {y, z})| ≥ f (n − 2) ≥ f (n) − 3, a contradiction. Hence G has no K3 -lobe.
As f (n − 3) ≥ f (n) − 5, a nearly identical argument demonstrates that G also
has no K4 -lobe.
58
We require a bit more terminology as we proceed. By an extended book we
mean a book with at most one pendant adjacent to each page (for instance, H0
and H1 are both extended books). The spine of an extended book is defined
analogously to that of a traditional book, and we refer to both a point joined
to the spine or a K2 -block as a page of an extended book. Additionally, by an
end-lobe of G we mean and end-block B of G− together with some or all of the
pendants in G that are adjacent to a vertex in B.
S1
S0
Figure 4.4: Exceptional K4 -type Lobes
Consider the graphs S0 and S1 depicted in Figure 4.4. If some maximal endlobe L is isomorphic to S1 , then by Lemma 4.2.6, there is no block of G that is
both isomorphic to K2 and incident to x, save for the one contained within L.
Consequently, we may replace L with a copy of H1 , where x is one of the vertices
on the spine of H1 . Similarly, if there is some maximal end-lobe L isomorphic
to S0 , then we can replace L with a copy of H0 , again with x playing the role of
one of the vertices on the spine of H0 . Either of these operations would result
in an S(C5 )-saturated graph with the same order and size as G that contains a
previously prohibited lobe. This implies that neither S0 nor S1 are lobes of G.
Suppose that some end-lobe L of G is isomorphic to an extended book with
59
t ≥ 3 pages and suppose as well that L is a maximal such end-lobe. By the
rearranging Lemma, we may assume that a vertex x on the spine of L is the
cut-vertex that separates L from the rest of G. If t > 3 and fewer than t − 1 of
these pages have pendants, then replace L with an extended book L0 having one
fewer page and one more pendant. The resulting graph is still S(C5 )-saturated,
has order n, and has fewer edges than G. Hence, if t > 3, we may assume
that either t − 1 or t of the pages of L contain pendant vertices. We may
modify G by deleting one of the leaves of L that contains a pendant and then
adding vertices x1 , x2 and the triangle x1 x2 x to G. This rearrangement results
in a S(C5 )-saturated graph with the same order and size as G that contains a
K3 -lobe, contradicting our above assumption.
Hence we may assume that any end-lobe of G that is isomorphic to an
extended book with at least three pages must have exactly three pages, at most
one of which contains a pendant vertex. Suppose first that G has some end-lobe
L isomorphic to a book with exactly three pages and one pendant edge. If we
let x be the cut-vertex of G that separates L from the rest of the graph and
G0 = G − (L − x) then G0 is still S(C5 )-saturated (by the Rearranging Lemma)
and |E(G0 )| = |E(G)| − 8. However, the induction hypothesis then implies that
|E(G0 )| ≥ f (n−5) which by Table 4.2 is at least f (n)−8, so that |E(G)| ≥ f (n),
a contradiction. Note as well that K4 with one vertex adjacent to a pendant
has order five and size eight, so that an identical argument also allows us to
eliminate this graph as an end-block of G.
It remains to consider only the possibility that every end-lobe of G is either
B3 or K4 with pendants attached to two vertices. However, in a manner similar
60
to our arguments above, if some end-lobe of L of G is B3 , then the fact that
f (n − 4) ≥ f (n) − 6 contradicts the assumption that |E(G)| < f (n). Similarly,
if L was a K4 with a pendant adjacent to exactly two of its vertices, then the
fact that f (n − 5) ≥ f (n) − 8 again facilitates a contradiction. This completes
the proof.
Next we determine sat(S(C6 ), n). The technique is quite similar to that
used to prove Theorem 4.2.8, so we will omit most of the detail in the interest
of concision.
Let Et be the the graph K2 ∨ (Kt ∪ K2 ). Some examples are depicted in
(Figure 4.5). We call Et an extended book and use the terms spine and pages in
a manner similar to our description of Bt .
Figure 4.5: The graphs E3 , E30 , and E300
Our next three Lemmas are analogous to Lemmas 4.2.5, 4.2.6 and 4.2.7.
Lemma 4.2.9. If G is a 2-connected S(C6 )-saturated graph of order n that is
not a clique, then n ≥ 6 and G ∼
= En−4 .
Lemma 4.2.10. Let G be an S(C6 )-saturated graph and B = xy a block of G
isomorphic to K2 . If x is a cut vertex of G and B 0 6= B is a block of G containing
x, then either B 0 ∼
= Et and x not on the spine of B 0 , or B 0 ∼
= K5 . Furthermore,
G · xy is also S(C6 )-saturated.
61
Lemma 4.2.11. Let G denote the family of graphs H with the following properties:
1. Every block of H is a clique of order at most 5 or a copy of Et with t ≥ 2,
and
2. For any K2 block B of H and block B 0 6= B of H, B ∩ B 0 6= ∅, either B 0
is a K5 -block of H or B 0 is an Et block of H with t ≥ 2 such that B ∩ B 0
is a non-spine vertex of B 0 .
Then, a graph G is S(C6 )-saturated of and only if G ∈ G.
We are now ready to determine sat(n, S(C6 ).
Theorem 4.2.12. For n ≥ 9, sat(n, S(C6 )) = g(n) = d 3n
e − 1.
2
Proof. Let Et0 be Et with a vertex pendant to each non-spine vertex, and let Et00
be Et0 with a single pendant vertex removed. See Figure 4.5b,c for examples.
For 9 ≤ n ≤ 17 we let the graphs E20 , . . . , E50 and E200 , . . . , E600 be our base cases
and we will proceed by induction on n.
We begin by addressing the base cases. for each value of n, 9 ≤ n ≤ 17,
assume G is an S(C6 )-saturated graph on n vertices of minimum size. If G were
2-connected then it must be the graph En−4 , which has more than g(n) edges.
Hence, in each case we can assume that G has connectivity 1.
Consider the x-lobes L of G where x is a cut-point. If L is composed entirely
of cliques and, possibly, copies of K5 with pendant vertices attached, then its
size is at least g(n). So, we can assume that G contains at least one x-lobe L
isomorphic to Et with a number, possibly zero, of pendants attached to unique
62
non-spine vertices. If a block of G is a K3 then this block can be replaced by
a page with a pendant added to L without increasing the size of G. Similarly,
if G contains a block isomorphic to K4 or K5 with pendants then the block can
be replaced by pages, all but possibly one attached to a pendant vertex, added
to L, with the resulting graph also S(C6 )-saturated but of smaller size. So, for
9 ≤ n ≤ 17, sat(n, S(C6 )) can be realized by one of E20 , . . . , E50 or E200 , . . . , E600 .
Let n > 9 and for all t, 9 ≤ t < n, sat(t, S(C6 )) = g(t). Now assume that
there is some S(C6 )-saturated graph G of order n with e(G) < g(n). We will
consider possible end-X−lobes of G.
Case 1: G has an end-lobe L isomorphic to K3 . By Lemma 4.2.11, G − (L − x) is
S(C6 )-saturated, n(G − (L − x)) = n − 2, and e(G − (L − x)) ≤ e(G) − 3 <
g(n) − 3 = g(n − 2), contradicting our inductive assumption.
Case 2: G has an end-lobe L isomorphic to K4 but none isomorphic to K3 . n(G −
(L−x)) = n−3, e(G−(L−x)) ≤ e(G)−6 < g(n)−6 = g(n−4) < g(n−3),
again contradicting the inductive assumption.
Case 3: G has an end-lobe isomorphic to K5 with i pendant vertices, 0 ≤ i ≤ 5,
but does contain an end-lobe like those in cases 1 or 2. n(G − (L − x)) =
n−(4+i), e(G−(L−x)) ≤ e(G)−(10+i) < g(n)−(10+i) < g(n−(4+i)),
once again contradicting the inductive assumption.
Case 4: G has end-lobes isomorphic to Et with i pendant vertices, 0 ≤ i ≤ (t + 2),
but does not contain an end-lobe like those in the previous cases. Let L be
the lobe with the least i among those with the least t. n(G − (L − x)) =
n − (t + 4 + i), e(G − (L − x)) ≤ e(G) − (6 + 2t + i) < g(n) − (6 + 2t + i) =
63
g(n−2)−(2t+i) < g(n−(t+4+i)), contradicting our inductive assumption
a final time.
4.2.3 General Bounds
The main result of this section is as follows.
Theorem 4.2.13. For t ≥ 9, there exists a constant c(t), tending to zero as t
tends to infinity, such that
5n
4
≤ sat(n, S(Ct ) ≤ ( 45 + c(t))n.
The following Lemma allows us to establish the lower bound in Theorem
4.2.13.
Lemma 4.2.14. If G is a 2-connected graph of order n ≥ 6 such that every
2-cut of G is complete, then e(G) ≥ 23 n.
Proof. If δ(G) ≥ 3 then e(G) ≥ 32 n. Thus we may assume that κ(G) = 2 and
that G contains at least one vertex of degree two. For convenience, through the
remainder of the proof we let P denote the property “every 2-cut is complete”.
We argue that if G has property P and x is a vertex of degree two in G,
then G − x also has property P. Suppose that there is a vertex x of degree two
in G and note that, as x is a vertex of degree two in G, its two neighbors form
a cutset and are therefore adjacent. If G − x does not have property P, then
G − x either is disconnected or contains an independent cutset B of size at most
two.
If G − x is disconnected, then the two neighbors of x belong to the same
component of G − x, and G is also disconnected, a contradiction. Assume then
64
that G − x is connected and that B is an independent 2-cut of G. This implies
that the two neighbors of x belong to the same B-lobe of G − x, and hence that
B is an independent 2-cut in G. This contradicts the assumption that G has
property P. Therefore deleting a vertex of degree two preserves property P.
Let G be a 2-connected graph with property P on at least six vertices. We
may inductively remove vertices of degree two from G until we have a graph
G0 that contains no vertices of degree two. Thus G0 has minimum degree 3 or
is a single edge. If G0 has minimum degree three, then m(G0 ) ≥ 23 n(G0 ), and
each vertex in G that is not in G0 contributes exactly two edges to the edge
count for G. Thus n(G) ≥ 23 n(G). If G0 = K2 , then m(G) = 2n(G) − 3. Thus
n(G) ≥ 32 n(G) if n(G) ≥ 6.
Theorem 4.2.15. Let t and n be integers such that t ≥ 6 and n is sufficiently
large. If G is a S(Ct )-saturated graph of order n, then m(G) ≥ 54 n.
Proof. Let n = |V (G)| and m = |E(G)|. If G has connectivity κ ≥ 3, then
δ(G) ≥ 3, and m(G) ≥ 23 n. Thus we may assume that κ(G) ≤ 2.
Let A = {x, y} be a minimal 2-cut in G. If x and y are not adjacent, then
G + xy contains a cycle of length at least t containing xy. Because A is a cutset
of size 2, all the vertices in the cycle are in one A-lobe of G. Thus there is an
xy-path P of length t − 1 in an A-lobe. Because A is a minimal cut, x and y
both have a neighbor in each component of G − A, so there is an xy-path in
each A-lobe of G. Joining one of these to P completes a cycle of length at least
t. It follows that every 2-cut in G contains an edge. Consequently, if κ(G) ≥ 2
then G has property P, and the result follows directly from Lemma 4.2.14. We
therefore assume that κ(G) = 1.
65
Define a big block of G to be a block which contains at least 4 vertices and a
small block to be a block with at most three vertices. Note that two small blocks
cannot be incident, as an edge joining these two blocks would not complete a
cycle of length at least 6. Also, any block with fewer than t vertices must be
complete.
Let ` be the number of leaves in G, consider the block decomposition of G− .
Suppose that that there are j large blocks in G− , and note that each large block
B contributes at least 3n(B)/2 edges to G0 by Lemma 4.2.14. The large blocks
are either incident or are joined by small blocks, and a vertex v in G− which is
not in a large block must be in a K3 -block B 0 . Since B 0 contains either one or
two cut vertices, the vertex v contributes either 3/2 (if B 0 has exactly one cute
vertex) or 3 (if B 0 has two cut vertices) to the edge count of G− . Thus e(G− ) is
at least
3(n−`)
.
2
Consequently, we know that e(G) is at least
3(n−`)
2
+ `. As no two small
blocks can be incident to the same cut vertex, we know that ` ≤ n/2 so it
follows that G has size at least (3/2)n − (1/4)n = (5/4)n.
We now show that sat(n, S(Ct )) gets arbitrarily close to the lower bound
given in Theorem 4.2.15 as t tends to infinity. Our strategy is to modify several
constructions developed by Clark, Entringer, and Shapiro in [12] and [13].
Definition 4.2.16. Let G be a 3-regular graph and let e ∈ E(G) and {v1 , v2 , . . . , vk } ⊆
V (G). The graph G(e) is obtained from G via the addition of a vertex z adjacent to the endpoints of e. The graph G(v1 , v2 , . . . , vk ) is obtained from G via
the execution of a ∆ − Y exchange at each vi We let G(v1 , v2 , . . . , vk , e) denote
G(e)(v1 , v2 , . . . , vk ). For some examples, see Figure 4.6.
66
v
v
e
Figure 4.6: A graph G and the modifications G(e) and G(v, e)
One of the best current estimates for sat(n, Ct ) comes from a construction
using Isaacs’ family of snarks[39].
Definition 4.2.17. Let m ≥ 3 be an odd integer. Let Jm denote the graph with
V (Jm ) = {vi : 0 ≤ i ≤ 4m − 1},
and
E(Jm ) = E0 ∪ E1 ∪ E2 ∪ E3
where
E0 = {{v4j , v4j+1 }, {v4j , v4j+2 }, {v4j , v4j+3 } : 0 ≤ j ≤ m − 1},
E1 = {{v4j+1 , v4j+7 } : 0 ≤ j ≤ m − 1},
E2 = {{v4j+2 , v4j+6 } : 0 ≤ j ≤ m − 1}, and
E3 = {{v4j+3 , v4j+5 } : 0 ≤ j ≤ m − 1}.
All subscripts are taken modulo 4m.
Definition 4.2.18. Let G1 , G2 , . . . , Gk be graphs, and let u1 ∈ G1 , u2 ∈
G2 , . . . , uk ∈ Gk be vertices. The graph resulting from a copy of G1 , copies
of Gi − ui for 2 ≤ i ≤ k, and all edges between u1 and the vertices in Gi − ui
67
that are adjacent to ui in Gi , 2 ≤ i ≤ k, is referred to as identification at
{u1 , u2 , . . . , uk }.
The following were developed in [1].
Definition 4.2.19. Let G be a graph. The graph H is a G-builder if H is
G-saturated and the graph resulting from identifying multiple copies of H at a
vertex v in H is also G-saturated. For a family F of graphs define an F-builder
similarly.
Using this notion, Barefoot et. al were able to show the following:
Theorem 4.2.20. For n ≥ 4m ≥ 12, m odd, sat(n, C4m ) ≤
10m−1
n
8m−2
+ 2m.
The C4m -builder H4m used to prove Theorem 4.2.20 is a copy of Jm with
the addition of pendant vertices to every vertex save v0 , as shown in Figure 4.7.
These builders are then associated at v0 to form the desired saturated graph of
small size.
1
3
0
0
2
11
8
9
10
6
4
5
7
Figure 4.7: The graph J3 and a C12 -builder based on it
This construction, at first, seems less than optimal in light of the results in
Table 4.1 which show that sat(Ct , n) approaches n as t grows large. However, of
68
interest here is the observation that any graphs G constructed from the builders
H4m via association at v0 are also S(C4m )-saturated, as each block in such a G
has order at most 4m and circumference at most 4m − 1. This observation is
key, and we will use modifications to Issacs’ snarks to construct S(Ct ) graphs
similarly.
A graph G of order n is maximally non-hamiltonian if G is not hamiltonian,
but for any pairs of nonadjacent vertices x and y in G, G + xy is hamiltonian.
Theorem 4.2.21.
1. Let e = {v0 , v2 }. For m ≥ 5 the graph Jm (e) is maxi-
mally nonhamiltonian.
2. For m ≥ 5, Jm (v2 ) is maximally nonhamiltonian.
3. For m ≥ 7, Jm (v4 , v14 ) is maximally nonhamiltonian.
4. For m ≥ 9, Jm (v2 , v14 , v26 ) is maximally nonhamiltonian.
5. For m ≥ 9, Jm (v14 , e), Jm (v14 , v26 , e), and Jm (v14 , v26 , v38 , e) are maximally
nonhamiltonian.
We combine the constructions from Theorems 4.2.20 and 4.2.21 to achieve
an upper bound for sat(n, S(Ct )) for all sufficiently large t. A graph G is
hamiltonian-connected if for every pair of vertices u, v ∈ G there is a hamiltonian path from u to v. G is maximally nonhamiltonian-connected if G is not
hamiltonian-connected but for any pair of nonadjacent vertices u and v in G,
G+uv is hamiltonian connected. The following Theorem from [40] will be useful.
Theorem 4.2.22. Let m ≥ 7 be an odd integer. Jm is maximally nonhamiltonianconnected.
69
For ease of reference we will group to the modified snarks in the following
way. As above, and for the remainder of this section, let e = v0 v2 and let z be
the vertex in Jm (e) − Jm .
J0m = {Jm , Jm (v2 ), Jm (v4 , v14 ), Jm (v2 , v14 , v26 )}
J1m = {Jm (v14 , e), Jm (v14 , v26 , e), Jm (v14 , v26 , v38 , e)}
Jm = J0m ∪ J1m
We combine the constructions from Theorems 4.2.20 and 4.2.21 to achieve an
upper bound for sat(n, S(Ct )) for all sufficiently large t.
Lemma 4.2.23. Let G be a 3-regular maximally nonhamiltonian graph. Let xy
be an edge of G such that either N (x) − y or N (y) − x are independent. Then xy
is on a cycle of length n(G) − 1 in G. Furthermore, for any edge e ∈ G, e 6= xy
such that G(e) is maximally nonhamiltonian, xy is on a cycle of length n(G) in
G(e).
Proof. Suppose xy is an edge in G with N (y) = {x, y1 , y2 } and y1 not adjacent to
y2 . Since G is maximally nonhamiltonian there is a path P of length n(G) − 1
between y1 and y2 . Since y has degree 3 and lies on P , and y1 yy2 is not a
subpath of P , the edge xy must be on P . Therefore, either P = y1 yxP1 y2 or
P = y1 P1 xyy2 for some path P1 in G, (see Figure 4.8). In the former case
yxP1 y2 y is an (n(G) − 1)-cycle in G, and in the latter the cycle y1 P1 xyy1 has
length n(G) − 1. In both cases the edge xy is on an (n(G) − 1)-cycle in G. If
G(e), xy 6= e, is maximally nonhamiltonian we can utilize a hamiltonian path in
G(e) of length n(G) between y1 and y2 to construct an n(G)-cycle containing
xy.
70
x
x
y
y
y1
y2
y1
y2
Figure 4.8: Every edge in a modified snark is on a long cycle
Lemma 4.2.24. Let m ≥ 9 be an odd integer and let J be a graph in Jm . Let
u, v, w be vertices in J, with u and v not necessarily distinct. There exists a
u − w path P1 and a v − w path P2 in J such that the total length of P1 and P2
is at least n(J).
Proof. Let P1 be a longest u − w path in J. Note that every edge in J, except
e = v0 v2 and the edges v0 z, zv2 created in Definition 4.2.16 if J ∈ J1m , satisfies
the requirements of Lemma 4.2.23 since {v2 , v14 , v26 } and {v14 , v26 , v38 } are independent sets of vertices in Jm for m ≥ 9. However, if J ∈ J1m the path v0 zv2 can
replace the edge e in an n(J) − 2 path to create an n(J) − 1 path in J. So, e is
the only edge in J not necessarily on a cycle of length n(J) − 1. If u is adjacent
to w and uw 6= e or J ∈ J0m then by Lemma 4.2.23 uw is on an n(J) − 1 cycle.
If uw = e and J ∈ J1m then uw is on an n(J) − 2 cycle in J. Otherwise, since
J is maximally nonhamiltonian there is a hamiltonian path between u and w.
Therefore, P1 has length at least n(J) − 2. Similarly P2 , a longest v − w path in
J, has length at least n(J) − 2, implying that the sum of the lengths of P1 and
P2 is at least 2n(J) − 4 ≥ n(J).
We now construct a family of S(Ct )-saturated graphs.
71
Theorem 4.2.25. Let m ≥ 9 be an odd integer and let J ∈ J1m . The graph
resulting from adjoining at most one pendant vertex to all vertices of J other
than v0 and v2 results in an S(Cn(J) )-saturated graph. Similarly, if H ∈ J0m then
the graph resulting from adjoining at most one pendant vertex to all vertices of
H results in an S(Cn(H) )-saturated graph.
Proof. Let J ∈ J1m . Since J is maximally nonhamiltonian by Theorem 4.2.21,
J is also S(Cn(J) )-saturated. Also, as the addition of pendants to J does not
increase the circumference of J, it suffices to prove that the graph J 0 obtained by
adding a pendant vertex to each vertex in V (J) − {v0 , v2 } is S(Cn(J) )-saturated.
Since J does not contain a subdivided Cn(J) , neither does J 0 . Because J
is S(Cn(J) )-saturated by Theorem 4.2.21, we need only consider the addition of
edges to J 0 with at least one endpoint in J 0 − J. Let r be a pendant vertex in J 0
with neighbor x. Let y be a neighbor of x distinct from r and first consider the
addition of the edge ry. The edge xy lies on an n(J) − 1 cycle in J 0 by Lemma
4.2.23, so replacing the xy with the path xry in J 0 + ry results in an n(J) cycle.
Next, we consider J 0 + rw where w is not adjacent to x. Note that if xw is
added to J 0 an n(J)-cycle C results by Theorem 4.2.21. If rw is added instead,
a cycle of length n(J) + 1 results from C by replacing the subpath wx with wrx.
Similarly, if r1 is another pendant vertex in J 0 with neighbor w not adjacent to
x, then the addition of edge rr1 results in a cycle with length 2 greater than the
n(J) cycle containing xw in J + xw. Finally, if r2 is a pendant vertex adjacent
to y ∈ N (x) then the cycle of length n(J) − 1 in J 0 that includes xy, assured by
Lemma 4.2.23, can be extended to a cycle of length n(J) + 1 by replacing the
edge xy with the path xrr1 y. Therefore, J 0 is S(Cn(J) )-saturated.
72
In a similar manner, we can create S(Cn(H) )-saturated graphs by adjoining
at most one pendant vertex to all vertices of H ∈ J0m other than v0 . Due to the
similarities, we omit the proof here.
Note that, for J ∈ J1m we do not add pendant vertices adjacent to either v0
or v2 . To see why, consider the case in which r is a pendant vertex adjacent to
v0 . Since both v0 and v2 are adjacent to z, the addition of edge rv2 does not
create a cycle any longer than any cycle in J that contains the path v0 zv2 . The
same is true for any vertex pendant to v2 . Since J 0 does not already contain an
n(J)-cycle, neither would J 0 + rv2 .
Lemmas 4.2.23 and 4.2.24 and Theorem 4.2.25 yield the following Corollary.
Corollary 4.2.26. Every graph G ∈ J is a S(Cn(G) )-builder.
Denote by Gv the graph obtained by adding a single pendant vertex adjacent
to each vertex of a graph G other than v. Let n, r and s be integers such that
0 ≤ r < n(G) − 1, n ≥ 1 and n = r + s(2n(G) − 1) + 1. Denote by Grv,s a graph
on n vertices composed of s copies of Gv and and a single copy of Grv identified
at v. By Corollary 4.2.26, if G is one of the possibly modified Jm ’s from above
then Grv,s is SCt saturated for t = n(G).
We now determine upper bounds on sat(n, S(Ct )) for sufficiently large t.
Theorem 4.2.27. Let m ≥ 9 be an odd integer.
sat(n, S(C4m )) ≤
10m − 1
n + 2m
8m − 2
10m + 1
+ 2m + 1
8m − 1
10m + 3
sat(n, S(C4m+2 )) ≤ n
+ 2m
8m + 2
sat(n, S(C4m+1 )) ≤ n
73
(4.1)
(4.2)
(4.3)
10m + 6
+ 2m + 2
8m + 3
10m + 9
sat(n, S(C4m+4 )) ≤ n
+ 2m + 7
8m + 6
10m + 11
+ 2m + 3
sat(n, S(C4m+5 )) ≤ n
8m + 7
10m + 14
sat(n, S(C4m+6 )) ≤ n
+ 2m + 3
8m + 10
10m + 16
sat(n, S(C4m+7 )) ≤ n
+ 2m + 4
8m + 11
sat(n, S(C4m+3 )) ≤ n
(4.4)
(4.5)
(4.6)
(4.7)
(4.8)
Proof. The proof of (4.1) is from [1], and we reproduce it here to demonstrate
how each equation in the Theorem is obtained. Let G be the graph obtained by
adding a pendant vertex to every vertex in Jm other than v0 . n(G−v0 ) = 8m−2,
n−1
so construct the graph G(4m, n) by associating s = d 8m−2
e copies of G at v0
and deleting r = 1 + s(8m − 2) − n pendants. Note that 0 ≤ r ≤ 8m − 3. By
Corollary 4.2.26 G(4m, n) is S(C4m )-saturated and has order n = 1+s(8m−2)−r
and size s(10m − 1) − r =
sat(n, S(C4m )) ≤
10m−1
n
8m−2
10m−1
(n
8m−2
− 1) +
2m+1
r
8m−2
<
10m−1
n
8m−2
+ 2m. Therefore
+ 2m.
To establish (4.2) let G be the graph obtained by adding a pendant vertex
to every vertex in Jm (e) other than v0 and v2 . Construct G(4m + 1, n) by
n−1
associating s = d 8m−1
e copies of G at v0 and deleting r = 1 + s(8m − 1) − n
pendants. By Corollary 4.2.26 G(4m+1, n) is S(C4m+1 )-saturated and has order
n = 1 + s(8m − 1) − r and size s(10m + 1) − r <
10m+1
n
8m−1
+ 2m + 1.
To establish (4.3) let G be the graph obtained by adding a pendant vertex
to every vertex in Jm (v2 ) other than v0 . Construct G(4m + 2, n) by associating
n−1
s = d 8m+2
e copies of G at v0 and deleting r = 1 + s(8m + 2) − n pendants.
By Corollary 4.2.26 G(4m + 2, n) is Sub(C4m+2 )-saturated and has order n =
1 + s(8m + 2) − r and size s(10m + 3) − r <
74
10m+3
8m+2
+ 2m.
To establish (4.4) let G be the graph obtained by adding a pendant vertex
to every vertex in Jm (v14 , e) other than v0 and v2 . Construct G(4m + 3, n) by
n−1
associating s = d 8m+3
e copies of G at v0 and deleting r = 1 + s(8m + 3) − n
pendants. is S(C4m+3 )-saturated and has order n = 1 + s(8m + 3) − r and size
s(10m + 6) − r <
10m+6
n
8m+3
+ 2m + 2.
To establish (4.5) let G be the graph obtained by adding a pendant vertex to
every vertex in Jm (v4 , v14 ) other than v0 . Construct G(4m + 4, n) by associating
n−1
e copies of G at v0 and deleting r = 1 + s(8m + 6) − n pendants.
s = d 8m+6
By Corollary 4.2.26 G(4m + 4, n) is Sub(C4m+4 )-saturated and has order n =
1 + s(8m + 6) − r and size s(10m + 9) − r <
10m+9
8m+6
+ 2m + 7.
To establish (4.6) let G be the graph obtained by adding a pendant vertex
to every vertex in Jm (v14 , v26 , e) other than v0 and v2 . Construct G(4m + 5, n)
n−1
e copies of G at v0 and deleting r = 1 + s(8m + 7) − n
by associating s = d 8m+7
pendants. is S(C4m+5 )-saturated and has order n = 1 + s(8m + 7) − r and size
s(10m + 11) − r <
10m+11
n
8m+7
+ 2m + 3.
To establish (4.7) let G be the graph obtained by adding a pendant vertex
to every vertex in Jm (v2 , v14 , v26 ) other than v0 . Construct G(4m + 6, n) by
n−1
e copies of G at v0 and deleting r = 1 + s(8m + 10) − n
associating s = d 8m+10
pendants. G(4m + 6, n) is Sub(C4m+6 )-saturated and has order n = 1 + s(8m +
10) − r and size s(10m + 14) − r <
10m+14
8m+10
+ 2m + 3.
To establish (4.8) let G be the graph obtained by adding a pendant vertex to
every vertex in Jm (v14 , v26 , v38 , e) other than v0 and v2 . Construct G(4m + 7, n)
n−1
by associating s = d 8m+11
e copies of G at v0 and deleting r = 1 + s(8m + 11) − n
pendants. G(4m + 7, n) is S(C4m+7 )-saturated and has order n = 1 + s(8m +
75
11) − r and size s(10m + 16) − r <
10m+16
n
8m+11
+ 2m + 4.
The reader should note that (4.5), and (4.7) may be improved slightly by
the addition of a single vertex pendant to v0 after associating the copies of
Jm (v4 , v14 ) and Jm (v4 , v14 , v26 ) respectively in the construction of G(t, n) but
this does not affect the asymptotics of our result.
4.3 Complete Graphs
In this section, we consider S(Kt )-saturated graphs. Since every subdivision
of K3 is a cycle, it follows that sat(n, K3 ) = sat(n, S(K3 ) = n − 1. The following Theorem of Dirac [15] is central to our characterization of S(K4 )-saturated
graphs.
Theorem 4.3.1. If δ(G) ≥ 3 then G contains a subdivision of K4 .
Lemma 4.3.1 implies that any S(K4 )-saturated graph has minimum degree
at most two and it is in fact an exercise in [58] to show that the maximum
number of edges in a graph of order n containing no member of S(K4 ) is 2n − 3.
We are able to prove a much stronger result characterizing the class of S(K4 )saturated graphs. A t-tree is any graph that can be obtained from a Kt by
repeatedly adding t-simplicial vertices.
Theorem 4.3.2. A graph G of order at least two is S(K4 )-saturated if and only
if G is a 2-tree.
Proof. We demonstrate that an arbitrary 2-tree G is S(K4 )-saturated by induction on |G|. The base cases |G| = 2 and |G| = 3 are immediate, as K2 and
76
K3 are vacuously S(K4 )-saturated. Suppose then that |G| ≥ 3 and let v be a
2-simplicial vertex in G and let N (v) = {x, y}. Then the graph G0 obtained
by removing v from G is a 2-tree, so by induction G0 is S(K4 )-saturated. The
addition of v to G0 cannot contain a subdivision of K4 , as such a subdivision
would necessarily contain xvy within some edge-path. Since xy ∈ G0 , replacing
xvy with xy would yield a subdivision of K4 within G0 , a contradiction to the
assertion that G0 is S(K4 )-saturated. It remains to show that for any vertex u in
G − N [v], G + uv contains a subdivision of K4 . Any 2-tree of order at least three
is 2-connected, so Menger’s Theorem implies that there are internally disjoint
paths from u to {x, y} in G0 . These paths, along with the edges xy, xv, yv and
uv for a subdivision of K4 in G + uv. Therefore G is S(K4 )-saturated.
To establish the necessity, suppose that G is a S(K4 )-saturated graph and
proceed by induction on |G|. The base case, |G| = 2 is immediate, so we suppose
that |G| ≥ 3. Clearly, G must be connected, so suppose first that G has a cut
vertex x and let y1 and y2 be neighbors of x lying in distinct components of
G − x. The K4 -subdivision K created by adding y1 y2 to G must have all of its
branch vertices in the same lobe of G, since any edge cut in K4 has at least
three edges. Without loss of generality, assume that the branch vertices K lie in
the component of G − x containing y1 . We know that y1 y2 must be an edge in
K, and therefore there is some edge-path P of K containing y1 , y2 and x in that
order. We can replace the segment of P from y1 to x with the edge yx, creating
a K4 subdivision in G, a contradiction.
We may therefore assume that G is 2-connected and therefore, by Lemma
4.3.1, that there is a vertex v in G with d(v) = δ(G) = 2. Let N (v) = {x, y}
77
and suppose that x and y are nonadjacent. Then there is a K4 subdivision K
in G + xy which contains xy and, as d(v) = 2, cannot contain v. However, then
we can replace the edge xy in K with the path xvy, creating a subdivision of
K4 in G, again a contradiction to the assumption that G is S(K4 )-saturated.
It now follows that G − v is also S(K4 )-saturated. Indeed, if the addition of
any edge u1 u2 to G, where u1 , u2 6= v, created a subdivision of K4 containing
v, then necessarily this subdivision would have some edge-path containing the
xvy. Since x and y are adjacent, G + u1 u2 also contains a subdivision of G that
omits v, implying the desired conclusion that G − v is also S(K4 )-saturated. By
induction, G − v is a 2-tree, and the result follows.
As any 2-tree of order n has 2n − 3 edges, we obtain the following Corollary.
Corollary 4.3.3. For n ≥ 2, ex(n, S(K4 )) = sat(n, S(K4 )) = sat(n, K4 ) =
2n − 3.
At this stage, it is tempting to conjecture that sat(n, S(Kt )) = sat(n, Kt )
for all t. Our next two constructions demonstrate that this is not the case in
general.
Theorem 4.3.4. Let t ≥ 5 be an odd integer and let n = d(t − 1) + r for d ≥ 2
and 0 ≤ r ≤ t − 2. Then
t−1
t−1
sat(n, S(Kt )) ≤ d
+
+ 2r
2
2
=(
t−2
+ o(1))n.
2
Proof. Suppose first that r = 0 and furthermore that d = 2. We construct
a S(Kt )-saturated graph Gt,n , starting with two copies of Kt−1 , A and B, with
78
V (A) = {a1 , . . . , at−1 ) and V (B) = {b1 , . . . , bt−1 }. Add a matching that connects
ai to bi for each i, and then remove the edges ai ai+1 , for i = 1, 3, . . . , t − 2 from
A. For d > 2, replace each edge ai bi with a path Pi = ai , a1i , . . . , aid−2 , bi . Then
for each k between 1 and d − 2 connect aki to akj for every j 6= i − 1, i + 1 (where
we consider ak0 = akt−1 and akt = ak1 ). If r > 0, we add vertices v1 , . . . , vr such
that each vi is adjacent to ai and a1i (or bi , if d = 2). For an example when t = 5
see Figure 4.9.
Figure 4.9: An S(K5 )-saturated graph of order 18. (d = 4, r = 2)
If r = 0 then Gt has only t − 1 vertices of degree at least t − 1, implying
that Gt,n cannot contain a subdivision of Kt . If r > 0, then we observe that
the addition of vi cannot create a subdivision of Kt if one was not previously
present, as this would imply that in such a subdivision, some ai would be a
branch vertex, and both ai bi and ai vi bi would lie within distinct edge-paths, an
impossibility. Thus, for all n, Gt,n contains no member of S(Gt,n ) as a subgraph.
It remains to show that for any pair of nonadjacent vertices u and v in
V (Gt,n ),that Gt,n ) + uv contains a subdivision of Kt . Assume that r = 0 and
suppose without loss of generality that u lies on P1 , that is u = a1 , b1 or ai1 for
some i. For each choice of v that follows, Gt,n + uv contains a Kt -subdivision
79
with branch vertices b1 , b2 . . . , bt−1 and u, using each edge between the vertices
of B as an edge-path. consequently, we will only describe the edge-paths in this
subdivision incident to u. By symmetry we may assume that if u = ai1 , then
either v = bj or v = akj where k ≥ i. In other words, referring to Figure 4.9,
we may assume that v lies “below” u. Note that this assumption implies that
u 6= b1 as the set B induces a clique.
If v lies in P1 , then v 6= ai+1
(or, if u = a1 , v 6= a11 ) and the edge-paths
1
i+1
incident to u will be uvP1 b1 , uai3 P3 a3 a2 P2 b2 , uai+1
1 a3 P3 a3 , uP1 a1 at−1 Pt−1 bt−1
and uaij Pj bj for 4 ≤ j ≤ t − 2. If v lies in P2 , then the edge-paths incident to u
are uP1 b1 , uvP2 b2 , uP1 a1 at−1 Pt−1 bt−1 and uaij Pj bj for 3 ≤ j ≤ t − 2. For v lying
on Pt−1 , we have the edge-paths from u being uP1 b1 , uP1 a1 at−1 a2 P2 b2 , uvPt−1 bt−1
and uaij Pj bj for 3 ≤ j ≤ t − 2. Finally, when v lies in Pj for 3 ≤ j ≤ t − 2,
the edge-paths incident to u are uP1 b1 , uvPj bj , uaij Pj aj a2 P2 b2 , uP1 a1 at−2 Pt−2 bt−2
and uaij Pj bj for 4 ≤ j ≤ t − 2. Some examples of the subdivisions described
above can be found in
We have therefore verified that Gt,n is S(Kt )-saturated when r = 0. For
r ≥ 1, we proceed by induction on r. As noted above, Gt,n contains no Kt subdivision, even when r > 0, so we need only consider the addition of edge uvi
to G. Let G0 = G − vi and consider the Kt -subdivision K formed by adding the
edge uai to G0 (which must exist by induction). Replacing the edge ai u in K
with ai vi u yields a Kt -subdivision in G + uvi , as desired.
Theorem 4.3.5. Let t ≥ 6 be an even integer and let n = d(t − 1) + r for d ≥ 2
80
u
u
v
v
u
v
u
v
Figure 4.10: Examples of K5 -subdivisions in G(5, 18) + uv. Open circles
represent the branch vertices of the subdivisions formed.
and 0 ≤ r ≤ t − 2. Then
t−1
t
sat(n, S(Kt )) ≤ d
+ + 2r − 2
2
2
=(
t−2
+ o(1))n.
2
Proof. As above, we construct an S(Kt )-saturated graph G(t, n) of the desired
size, starting with d = 2 and r = 0. We again begin with two copies of Kt−1 ,
A and B, with V (A) = {a1 , . . . , at−1 ) and V (B) = {b1 , . . . , bt−1 } joined by the
edges ai bi for 1 ≤ i ≤ t − 1. we then remove the edges a1 a2 , a2 a3 and a3 a1 as well
as the edges ai ai+1 for i = 4, 6, . . . , t − 2 and add the edges a1 b2 , a2 b3 and a3 b1 .
For d > 2, replace the edges ai bi with a path Pi = ai , a1i , . . . , ad−2
, bi . Then for
i
each k between 1 and d − 2 connect aki to akj for every j 6= i − 1, i + 1 (where we
consider ak0 = akt−1 and akt = ak1 ). If r > 0, we add vertices v1 , . . . , vr such that
each vi is adjacent to ai and a1i (or bi , if d = 2). For an example when t = 6, see
Figure 4.11.
81
Figure 4.11: G(6, 21), an S(K6 )-saturated graph of order 21. (d = 4, r = 1)
The proof that G(t, n) contains no subdivision of Kt is identical to the
argument in the case where t is odd, and hence we have elected not to repeat it
here.
It remains to verify that for any nonadjacent u and v in G = G(t, n), G + uv
contains a Kt -subdivision. As above, we assume that r = 0. In the interest
of concision, we consider only the case where u lies on P1 , as the remaining
possibilities are handled similarly. As above, by symmetry we may also assume
that v lies at or below the same ”vertical” level as u in the drawing depicted in
Figure 4.11. Some examples of the subdivisions created for various choices of u
and v are depicted in Figure 4.12.
For each choice of u and v that follows, the vertices of B will all be branch
vertices of the subdivision formed, with the edges connecting vertices of B serving as edge-paths. The remaining branch vertex will be either u or v, depending
on the situation, and hence we only describe the edge-paths incident to this
vertex. For the remainder of the proof, given a u (respectively v on some Pi ,
we let uk (resp. vk ), k 6= i, denote the neighbor of u (v) on Pk , provided such a
neighbor exists.
82
u
u
v
v
u
u
v
v
Figure 4.12: Examples of K6 -subdivisions in G(6, 21) + uv. Open circles
represent the branch vertices of the subdivisions formed.
Suppose first that u lies on P1 , so that u = ai1 for some 0 ≤ i ≤ d−3. If v also
lies in P1 , then v 6= ai+1
and u is a branch vertex. and the edge paths incident
1
i+1 i+1
i
to u are uvP1 b1 , uP1 a1 b2 , uai+1
1 a3 at−1 Pt−1 bt−1 , and uak Pk bk for 3 ≤ k ≤ t − 2.
If v lies on P2 , we must handle the case where v = b2 differently from the
other possible choices of v. If v = b2 , then u 6= a1 will be the additional branch
vertex in the Kt -subdivision including uv, with the edge-paths incident to u
being uP1 b1 , ub2 , uP1 a1 at−1 Pt−1 bt−1 and uuk Pk bk for 3 ≤ k ≤ t − 2. If v 6= b2 ,
then v is a branch vertex in a Kt -subdivision, and the edge-paths incident to v
are vuP1 b1 , vP2 b2 , vP2 a2 b3 and vvk Pk bk for 4 ≤ k ≤ t − 1.
For v lying on P3 , and v 6= b3 , we have v as the branch vertex of the
Kt -subdivision in G + uv, with vP3 a3 b1 , vuP1 a1 b2 , vb3 , vv1 v4 P4 b4 and vvk Pk bk ,
5 ≤ k ≤ t − 1 comprising the edge-paths incident to v.
83
When v ∈ Pj , for 4 ≤ j ≤ t − 4, then u is the additional branch vertex, u 6= uj , and the u − bi edge-paths for i 6= 3 are uP1 b1 , uP1 a1 b2 , uvPj bj ,
uuj u2 ut−1 Pt−1 bt−1 and uuk Pk bk for 5 ≤ k ≤ t − 2. The u − b3 edge-path depends
on whether u = a1 , in which case v 6= aj and the desired edge-path is uaj a3 P3 b3 .
If u 6= a1 , then the desired edge path is simply uu3 P3 b3 .
84
5. Edge-Critical G, H Colorings
5.1 Introduction
Let G, H be simple graphs, not necessarily distinct, each containing at least
one edge and not both containing exactly 1 edge. We say that a coloring C of
a simple graph F is an edge-critical G,H coloring if it contains no red G nor
blue H, but changing a red edge to blue or a blue edge to red eliminates this
property. We denote by CSat(G,H) the set of colored graphs that are edgecritical G, H, and by csat(G,H) the fewest number of edges over all elements of
CSat(G, H). As a simple example consider G = H = P3 , the path on 3 vertices.
Let F also be P3 , and C the coloring of F consisting of one red edge and the
other blue. This coloring of F has no monochromatic P3 , but reversing the blue
edge results in a red P3 and reversing the red edge results in a blue P3 . So,
(F, C) ∈ CSat(P3 , P3 ). Every graph in CSat(G, H) must contain at least one of
G, H as an uncolored subgraph, so F has the fewest number of edges possible for
a graph with an edge-critical P3 , P3 coloring. In fact, it is the unique smallest
such graph and the coloring C is the unique edge-critical P3 , P3 coloring on F
up to switching the colors. Therefore, csat(P3 , P3 ) = 2.
Note that CSat(G, H) need not be finite. In fact, a minimum proper edge
coloring of any disjoint union of any number of even cycles and a linear forest
is also edge-critical P3 , P3 , (see Fig. 5.1). If F has a vertex of degree at least
3 then every 2-coloring of F contains a monochromatic P3 . So, CSat(P3 , P3 )
is precisely the set of all disjoint unions of even cycles and linear forests with
proper edge colorings.
85
Figure 5.1: An edge-critical P3 , P3 coloring of a graph
Given any pair of graphs G and H, the disjoint union of any collection of
graphs in CSat(G, H) is also in CSat(G, H). Therefore, if CSat(G, H) is not
empty, then it contains infinitely many colored graphs.
Theorem 5.1.1. If G is a graph with at least 2 edges then CSat(G, K2 ), CSat(K2 , G)
are non-empty and csat(G, K2 ) = csat(K2 , G) = 1.
Proof. Consider the graph K2 colored red.
This is an edge-critical G, K2
coloring. Similarly, a blue K2 is an edge-critical K2 , G coloring. Since every element of CSat(G, K2 ) and CSat(K2 , G) must contain at least one edge,
csat(G, K2 ) = csat(K2 , G) = 1.
5.2 Kneser Graphs
Let i, j be integers no less than 3. We will show that CSat(Ki , Kj ) is nonempty.
86
1,2
4,5
3,5
3,4
1,5
2,4
1,3
2,3
2,5
1,4
Figure 5.2: The Petersen Graph KG(5, 2)
Definition 5.2.1. The Kneser Graph KG(n, k) for n ≥ k > 0 has as vertices
the k-element subsets of {1, 2, . . . , n} = [n]. A pair of vertices is adjacent if and
only if their associated sets are disjoint.
The Petersen Graph, (Fig. 5.2), is isomorphic to KG(5, 2). It is easily seen
that KG(n, k) has nk vertices and is n−k
-regular.
k
Theorem 5.2.2. Let m, p ≥ 2 be integers. KG(mp − 1, p) is Km -saturated.
Proof. We must first show that KG(mp − 1, p) contains no clique of order m.
Any maximal collection of pairwise disjoint p element subsets of [mp−1] contains
m − 1 sets, with p − 1 elements remaining. Since every clique in KG(mp − 1, p)
is associated with such a collection we can see that no clique has more than
m − 1 vertices. Now, we must show that the addition of any edge between
non-adjacent vertices of KG(mp − 1, p) creates a Km . If u, v are non-adjacent
vertices of KG(mp − 1, p) then they must be p-element sets of the form u =
{n0 , n1 , . . . , np−1 }, v = {n0 , np , . . . n2p−2 } for {ni }02p−2 ⊆ [mp − 1], not necessarily
87
distinct. There are at least mp − 1 − (2p − 1) = (m − 2)p elements remaining
unused in [mp − 1], and hence there is a collection of m − 2 pairwise disjoint p
element sets also disjoint from {ni }02p−2 . This collection corresponds to an m − 2
clique in the shared neighborhood of u and v, N (u)∩N (v). Joining u to v creates
an m clique in KG(mp − 1, p). Therefore, KG(mp − 1, p) is Km -saturated.
We use Kneser graphs of this form to create a graph Fi,j with edge-critical
Ki , Kj coloring C.
Let Fi,j be the graph with vertices of the form (r, b), where r is a 2 element
subset of [2i−1] and b is a 2 element subset of [2j −1]. A red edge exists between
(r, b) and (ρ, β) when r and ρ are disjoint but b and β are not. Similarly, a blue
edge exists between (r, b) and (ρ, β) when b and β are disjoint but r and ρ are
not. Note that Fi,j is a coloring of the complement of the modular product of
KG(2i − 1, 2) and KG(2j − 1, 2).
Lemma 5.2.3. C is an edge-critical Ki , Kj coloring of Fi,j .
Proof. A red clique in Fi,j must be associated with pairwise disjoint 2 element
sets of [2i − 1] and pairwise intersecting sets of [2j − 1], so there can be no
red clique in Fi,j larger than the largest clique in KG(2i − 1). If {sk }i−1
is a
1
collection of pairwise disjoint subsets of [2i − 1] then {(si , {1, 2})}i−1
induces a
1
red i − 1 clique, and thus a largest red clique, in Fi,j . Similary, a largest blue
clique in Fi,j has order j − 1. We must show that reversing the color of any edge
in Fi,j creates a red Ki or a blue Kj . Consider a blue edge in Fi,j . It must be
of the form ({r1 , r2 }, {b1 , b2 }) ∼ ({r1 , r3 }, {b3 , b4 }) where r1 , r2 , r3 are elements
of [2i − 1] and b1 , b2 , b3 , b4 are distinct elements of [2j − 1]. Let {sk }i−2
be a
1
88
collection of pairwise disjoint 2 element sets of [2i − 1]\{r1 , r2 , r3 }, which itself
contains at least 2i − 4 elements. The collection {(sk , {b1 , b3 })}1i−2 corresponds
to an i − 2 red clique in the shared red neighborhood of ({r1 , r2 }, {b1 , b2 }) and
({r1 , r3 }, {b3 , b4 }) in Fi,j . So, if the edge ({r1 , r2 }, {b1 , b2 }) ∼ ({r1 , r3 }, {b3 , b4 })
becomes red then we get a red Ki in Fi,j . A similar argument shows that a blue
Kj arises when any red edge is turned blue.
Theorem 5.2.4. CSat(Ki , Kj ) is non-empty.
Proof. Fi,j with coloring C as defined above is in CSat(Ki , Kj ).
5.3 Some Results
Note that if G, H are simple graphs such that the graph F with coloring C is
in CSat(G, H) then F with coloring C, the complement of C, is in CSat(H, G).
Therefore, we need only consider one of the two sets.
Theorem 5.3.1. csat(K3 , K3 ) = 10
Proof. We begin by proving the lower bound. Say F with coloring C, denoted
(F, C), realizes csat(K3 , K3 ). Without loss of generality assume that e(B) ≥
e(R), where B and R are the blue and red induced subgraphs, respectively, of
(F, C). (F, C) contains a triangle with two blue edges and one red. Label the
blue edges b1 , b2 and the red edge r1 as in Fig. 5.3. The endpoints of b1 must
share a red neighbor, which must be distinct from the vertices of the original
triangle. So, there are red edges r2 , r3 with a shared endpoint such that both
are incident with b1 . Similarly there are red edges r4 , r5 both incident with
b2 and sharing an endpoint. These must be distinct from r1 , r2 , r3 else (F, C)
89
b2
b1
r1
r2
Figure 5.3:
graph
r5
r3
r4
A colored subgraph of every edge-critical K3 , K3 coloring of a
contains a red K3 . So, e(R) ≥ 5. By a symmetric argument e(B) ≥ 5 as
well, hence e(F ) ≥ 10. C5 , which is isomorphic to its complement, yields an
edge critical K3 , K3 coloring of K5 . K5 achieves the lower bound, and hence
csat(K3 , K3 ) = 10.
Definition 5.3.2. Let q be a prime power, q ≡ 1 (mod 4). The Paley graph
of order q has vertex set [q] and edge ij if and only if (i − j) is a square in the
finite field GF (q).
C5 is the Paley graph of order 5. The Paley graphs of order 9 and 13 each
yield an edge-critical K4 , K4 coloring of the complete graph on their vertices.
However, this can only be verified for finitely many Paley graphs, since their
clique size is not known in general.
Let m ≥ 3 be an integer. We can determine csat(P3 , Km ) with the following
construction.
Theorem 5.3.3. csat(P3 , Km ) =
m
2
+ m(m − 3)
Proof. Color all but one edge of the complete graph Km blue, with the remaining
edge colored red, (Fig. 5.4). This colored graph must be a subgraph of every
90
Figure 5.4: Minimal edge-critical P3 , K6 coloring of F
element of CSat(P3 , Km ). Since all but at most one vertex in this coloring must
be incident to a red edge, add a single pendant red edge to m−3 of the remaining
m − 2 vertices. Since every red edge must complete a blue Km , every endpoint
of a red edge must have at least m − 1 neighbors along blue edges. So, add
a blue edge from each pendant vertex to the remaining m − 1 vertices of the
original complete graph. Denote by F the resulting underlying graph, and the
coloring C. Then, (F, C) ∈ CSat(P3 , Km ) and F has the minimum number of
edges. So, csat(P3 , Km ) = m2 + m(m − 3).
In fact, this construction can be used to show the existence of edge-critical
P3 , G and edge-critical P4 , G colorings of graphs for any graph G.
Theorem 5.3.4. Let G be a graph on n vertices with minimum degree δ. Then,
CSat(P4 , G) and CSat(P3 , G) are non-empty. Furthermore, csat(P4 , G) ≤
n−1
n−1
+
(n
−
1)δ
and
csat(P
,
G)
≤
+ (n − 2)δ
3
2
2
91
Figure 5.5: An edge-critical G, P3 coloring of a graph H in which n(G) =
7, δ(G) = 3
Proof. Consider the graph H consisting of 2n−2 vertices {v0 , v1 , . . . , vn−2 , u0 , u1 , . . . , un−2 }.
Include all edges of the form {vi , vj }n−2
i=0 and color them blue. For every integer i, 0 ≤ i ≤ n − 2 add the blue edges {ui , vj }i+δ
j=i+1 and the red edge {ui , vi },
with addition modulo (n − 1). There is no collection of n vertices with the
property that every vertex has degree at least δ in the induced blue subgraph
and hence H with this coloring has no blue G. The induced red subgraph is
simply a matching and thus contains no red P3 . In fact, the red subgraph is
a P4 -saturated graph on the vertices of H, so if any blue edge is turned red
it is clear that a red P4 arises. If a red edge, say {ui , vi }, is turned blue then
the induced subgraph on the vertices {v0 , v1 , . . . , vn−2 , ui } is blue and contains
a copy of G. Therefore, H with this coloring is in CSat(G, P4 ). It is easily seen
that H contains n−1
+ (n − 1)δ edges.
2
Let H 0 be the graph H with the vertex un−2 removed, and color it the same
as H, (Fig. 5.5). Similar to above the red subgraph of this coloring of H 0 is a
P3 -saturated graph on the vertices of H 0 , and hence CSat(P3 , G) is non-empty
and is bounded above by csat(P3 , G) ≤ n−1
+ (n − 2)δ.
2
92
Figure 5.6: An edge-critical G, G coloring for κ0 (G) = 1
Following are some existence results for diagonal cases.
Theorem 5.3.5. CSat(G, G) is non-empty if G has at least one of the following
two properties.
• G has edge connectivity 1.
• G can be realized by a simple graph H with a single edge subdivided twice.
Proof. To prove the sufficiency of the first condition say G has a cut edge e such
that G − e contains two connected components of order k ≤ j. Consider the
colored graph determined by the cartesian product Kj Kj , with the first red
and the second blue, as in Fig. 5.6.
Since each blue (red) component in the coloring has only j vertices, there
is no blue (red) G. However, if any red (blue) edge changes color we get a pair
of blue (red) j cliques joined by a single edge. The resulting coloring contains
G as a blue (red) subgraph. Therefore, the original coloring of Kj Kj is an
edge-critical G, G coloring.
93
Kj
Kj
Figure 5.7: G has a pair of adjacent degree 2 vertices not on a triangle
Now assume that G satisfies the second condition but not the first, n(G) = n.
Therefore, G has no vertex of degree 1. Let F be the graph consisting of a pair
of n − 1 cliques {v0 , v1 , . . . , vn−2 } and {u0 , u1 , . . . , un−2 }. Color the edges of the
first red and those of the second blue. Add a red edge from vi to ui and a blue
edge from vi to ui+1 for each i, with addition modulo n − 1. Since G has no
pendant vertices there is no monochromatic G in F with coloring C. If we turn
any blue edge of the form ui uj red then vi ui uj vj is our subdivided edge in a
red subgraph isomorphic to G, (Fig. 5.7). If we turn any blue edge of the form
vi ui+1 red then we get every graph on n vertices with a vertex of degree 1 or
2 as a red subgraph of F . Therefore, G is isomorphic to a red subgraph of the
resulting coloring of F . Similarly, turning any red edge of the form vi vj or of
the form vi ui blue results in a blue subgraph isomorphic to G.
Note that the first case can be extended to show that CSat(G, H) is nonempty if both G and H have edge connectivity 1, even if they are not isomorphic,
by using red cliques of one order and blue cliques of another.
94
We can also show constructively that CSat(G, Km ) is non-empty for any
graph G with edge connectivity 1. We restate Def. 1.3.4 here for convenience.
Definition 5.3.6. Let n, r be integers, n > r. The Turán Graph T (n, r) is the
complete multipartite graph on n vertices composed of r parts, in which every
part has order d nr e or b nr c. This is the unique largest (edge number) Kr+1 -free
graph on n vertices.
Theorem 5.3.7. Let G have edge connectivity 1, and let m be an integer greater
than 2. Then, CSat(G, Km ) is non-empty.
Proof. Say G has a cut edge e such that G−e contains two connected components
of order k ≤ j. Consider the complete graph Kj(m−1) consisting of a blue Turán
graph T (j, m − 1) and the remaining edges red. This coloring of Kj(m−1) has no
blue Km and no red G, but altering the color of any single edge eliminates this
property. Therefore, CSat(G, Km ) is non-empty.
95
6. Conclusion and Futher Directions
6.1 Oriented Graph Saturation
Saturated simple graphs have been studied extensively. There has been a
tremendous amount of interest in their structure, beginning with the work of
Mantel [46] and Turán [56] with regard to maximum size and Erdös, Hajnal, and
Moon’s work [21] on saturated graphs of minimum size. While directed graphs
have also received a great deal of attention, the intersection of these two topics
has thus far been very limited in scope, and what attention it has received has
been restricted to orientations of graphs containing multiple edges [20]. There
have been no results on oriented simple graph saturation prior to this thesis.
A possible reason for this lack of attention to oriented graph saturation is the
lack of assurance that the parameters sat(n, D) and ex(n, D) are well-defined
for all oriented graphs D. We have resolved this issue in Theorem 2.3.4, and
thus have made oriented graph saturation a viable field of study. We have also
seen in Chapter 2 that the saturation number of D can be, but is not necessarily, related to the saturation number of its underlying graph u(D). While
Bollobás demonstrates [2] that for integers n ≥ m the unique graph that realizes sat(n, Km ) is the smallest strongly Km -saturated graph on n vertices, this
is not necessarily the case for all graphs in general. Since a D-saturated oriented
graph F has the property that u(F ) is strongly u(D)-saturated, a solution to
the following problem will potentially resolve sat(n, D) for a number of oriented
graphs.
96
Problem 6.1.1. For which graphs G is sat(n, G) the fewest number of edges in
a strongly G-saturated graph on n vertices?
Clearly it would be advantageous to determine sat(n, D) for other families
of oriented graphs, including non-transitive tournaments, oriented trees, and
orientations of paths. While we have established that n ≤ sat(n, P~m ) ≤ m−2
+
2
2(n − m + 2), we have no reason to believe that either of these bounds will be
met. In fact, it is entirely possible that a construction similar those presented in
Chapter 4 will realize sat(n, P~m ). This is definitely a direction for future work
in this area.
Another interesting opportunity for extending oriented graph saturation is
the study of symmetric digraphs.
Definition 6.1.2. A digraph D is symmetric if for any arc xy in D the arc
yx is also in D. For a simple graph G let s(G) be the symmetric graph on the
vertices of G in which xy, yx are arcs in s(G) if and only if xy is an edge in G.
For any symmetric digraph F let f (F ) =
1
|A(F )|,
2
so that f represents
the number of unique pairs of adjacent vertices. Let D be an oriented graph.
If s(G) is a D-saturated digraph for some graph G then G is u(D)-saturated.
The converse is not necessarily true, as can be seen in Figure 6.1. The graph
H = Km ∪ K1 is K1,m -saturated, but s(H) is not K1,(m,0) -saturated.
Extend the definition of sat to symmetric digraphs in the following way. For
an oriented graph D let sats (n, D) = min{f (F )} over all D-saturated symmetric
digraphs F on n vertices.
Problem 6.1.3. How does sats (n, D) relate to sat(n, D) and sat(n, u(D))?
97
y
x
Figure 6.1: The addition of arc xy to s(K4 ∪ K1 ) does not create a K1,(4,0)
It is clear that sats (n, D) ≥ sat(n, u(D). When is equality achieved?
6.2 Maximal Non-Interval Graphs and Bigraphs
Identification of interval graphs and bigraphs has been a topic of interest
in graph theory for some time. Since Lekkerkerker and Boland determined a
simple forbidden subgraph characterization of interval graphs in 1962 [43], it
has been assumed that a similar characterization could be found for interval
bigraphs. In fact [reference(s) for conjectures of complete forbidden subgraph
characterizations, but were shown wrong]. Early in their study [35] interval
bigraphs were thought to be similar enough to interval graphs that a simple
translation of the forbidden family of asteroidal triples and induced cycles of
length at least four into the family of asteroidal triples of edges and induced
cycles of length at least six would suffice for determining whether or not a graph
was an interval bigraph. This family was shown to be insufficient in [47], and we
are currently left with the graphs in Theorems 3.3.4 and 3.3.5. This family has
not yet proven to be complete, and graphs like those in [37] may in fact confirm
that it is not.
98
Given a family G of graphs characterized by a family F of forbidden induced
subgraphs it is not a surprise that any graph that is edge-maximal G-free is
the join of a graph in F with a complete graph, as we saw in Theorem 3.2.2.
What is interesting, however, is how similar the families in Theorem 3.2.2 and
Conjecture 3.3.15 are in light of the differences between the forbidden subgraphs
associated with interval graphs and interval bigraphs. It is of course our hope
that any new developments in the identification of interval bigraphs will support
Conjecture 3.3.15.
6.3 Subdivided graph saturation
In [30] Gould, Luczak, and Schmitt improve upon a previous upper bound
[1] on sat(n, Ct ), demonstrating that for n, t sufficiently large sat(n, Ct ) ≤ (1 +
2
2
)n + 5t4
t−2
2
2
when t is even and sat(n, Ct ) ≤ (1 + t−3
)n + 5t4 when t is odd. While
these values are lower than our results from section 4.2, it should be noted that
the Ct -saturated graphs presented in these papers include cycles much longer
than Ct . In fact, since Ct ∈ S(Ct ), we have effectively placed a lower bound on
the number of edges in any Ct -saturated graph with circumference at most t − 1,
and demonstrated that all cycle saturated graphs with fewer edges include long
cycles.
This result gives rise to an interesting question.
Problem 6.3.1. Let n, g ≥ 3 be integers, G a graph. The circumference of
a graph is the length of its longest cycle. Determine sat(n, G, g), the minimum
number of edges in a G-saturated graph on n vertices with circumference at most
g, in particular when G is a cycle.
99
In section 4.3 we gave constructions that place upper bounds on sat(n, S(Kt )).
This bound is on the order of 2t n. These constructions have the property that
the addition of any edge between nonadjacent vertices x, y creates a subdivided
Kt with x or y a branch vertex. Hence these graphs have high minimum degree.
We have no reason to assume that all S(Kt )-saturated graphs must have this
property.
Problem 6.3.2. Let n ≥ t ≥ 7 be integers. Is there a S(Kt )-saturated graph G
on n vertices with nonadjacent vertices x, y such that in G + xy neither x nor y
is a branch vertex of a subdivided Kt ?
If the answer to problem 6.3.2 is yes, then there is a good chance that
sat(n, S(Kt )) is much lower than the upper bound we give in Theorems 4.3.4
and 4.3.5.
6.4 Edge-Critical G, H Colorings
With regard to edge-critical G, H colorings examined in Chapter 5, note that
we have not yet determined an upper bound on csat(G, H) for all pairs of graphs
G, H. In fact, we have not even shown existence of edge-critical G, H colored
graphs for all such pairs. It is possible that a variation on the Kneser graph
XOR construction in section 5.2 can be used to find a graph in CSat(G, H), but
we have yet to find such a generalization. Sparse graphs, such as paths, have
proven difficult to avoid in similar constructions using various graph products
and Kneser-like graphs.
A similar parameter from Ramsey-Turán theory examines non-complete
graphs with no edge colorings that avoid monochromatic forbidden subgraphs.
100
Definition 6.4.1. Let G, H be graphs, and let F be the family of graphs with
the property that any 2 edge coloring of any graph in F contains a red G or a
blue H. A graph in F is Ramsey minimal if no proper subgraph is in F. Let
rmin(G, H) denote the fewest number of edges over all graphs in F.
We have yet to find a pair G, H of graphs for which the following conjecture
does not hold.
Conjecture 6.4.2. For all non-trivial graphs G, H the family CSat(G, H) is
non-empty. Furthermore, csat(G, H) ≤ rmin(G, H) − 1.
There are potentially alterations to the XOR method that we have not
considered with regard to csat, and the method itself may also prove useful in
the related area of rmin.
101
REFERENCES
[1] C. Barefoot, L. Clark, R. Entringer, T. Porter, L. Székely, Z. Tuza, Cyclesaturated graphs of minimum size, Discrete Math. 150 (1996), 31-48
[2] B. Bollobás, Extremal Graph Theory, Academic Press Inc. (London) Ltd.
(1978)
[3] B. Bollobás, On generalized graphs, Acta Math. Acad. Sci. Hungar 16
(1965), 447-452
[4] J. A. Bondy, Variations on the Hamiltonian theme, Canad. Math. Bull., 15
(1972), 57-62
[5] F. Bonomo, G. Duran, L. Grippo, M. Safe, Partial Characterizations of
Circular-Arc Graphs, J. Graph Theory 61 (2009), 289-306
[6] D. Brown, S. Flink, J.R. Lungren, Characterizations of interval bigraphs
and unit interval bigraphs, Congr. Numer. 157 (2002), 79-93
[7] D. Brown, J.R. Lundgren, Relationships among classes of interval bigraphs
(0, 1)-matrices and circular arc graphs, Congr. Numer. 166 (2004), 97-123
[8] W. G. Brown, M. Simonovits, Digraph extremal problems, hypergraph extremal problems, and the densities of graph structures, Discrete Math. 48
(1984), 147-162
[9] Y. C. Chen, Minimum C5 -saturated graphs, J. Graph Theory 61(2) (2009),
111-126
[10] V. Chvátal, P. Erdős, A note on hamiltonian circuits, Discrete Math. (1973),
2
[11] L. Clark, R. Crane, R. Entringer, H. Shapiro, On smallest maximally nonHamiltonian graphs, Congr. Numer. 53 (1986), 215-220
[12] L. Clark, R. Entringer, Smallest Maximally Nonhamiltonian Graphs, Period. Math. Hungar. 14 (1983), 57-68
[13] L. Clark, R. Entringer, H. Shapiro, Smallest Maximallly Nonhamiltonian
Graphs II, Graphs Combin. 8 no. 3 (1992), 225-231
102
[14] H. S. M. Coxeter, A systematic notation for the Coxeter graph, C. R. Math.
Rep. Acad. Sci. Canada, 3(6) (1981) 329-332
[15] G. A. Dirac, A property of 4-chromatic graphs and some remarks on critical
graphs, J. Lond. Math. Soc. 27 (1952), 85-92
[16] G. A. Dirac, Some Theorems on abstract graphs, Proc. London Math. Soc.
2 (1952), 69-81
[17] G. A. Dirac, Homomorphism theorems for graphs, Math. Ann. 153 (1964),
69-80
[18] A. Duffus, D. Hanson, Minimal k-saturated and color critical graphs of
prescribed minimum degree, J. Graph Theory 10(1) (1986), 55-67
[19] J. Eckhoff, Extremal Interval Graphs, J. Graph Theory 17 (1993), 117-127
[20] P. Erdös, On some extremal problems on r-graphs, Discrete Math 1 (1971),
1-6
[21] P. Erdös, A. Hajnal and J.W. Moon, A problem in graph theory, Amer.
Math. Monthly 71 (1964), 1107-1110
[22] A. Dudek, G. Katona, A. Wojda, Hamiltonian path saturated graphs with
small size, Discrete Appl. Math 154(9) (2006), 1372-1379
[23] J. Faudree, R. Faudree, J. Schmitt, Survey of graph saturation numbers,
Personal communication (2010)
[24] R. Faudree, M. Ferrara, R. Gould, M. Jacobson, tKp -saturated graphs of
minimum size, Discrete Math. 309 no. 19 (2009), 5870-5876
[25] M. Ferrara, A. Harris, M. Jacobson, The Game of F-Saturator, Discrete
Appl. Math. 3(6) (2010) 189-197
[26] D. Fisher, K. Fraughnaugh, L. Langley, On C5 saturated graphs with minimum size, Congr. Numer. 112 (1995), 45-48
[27] S. Foldes and P. L. Hammer, Split Graphs having Dilworth number two,
Canad. J. Math. 39 (1977), 666-672
[28] M. Frick, J. Singleton, Lower bound for the size of maximal nontraceable
graphs, Electron. J. Graph Theory 18(1) (2005), 9
103
[29] Gallant, Gunther, Hartnell, Rall, A Game of Edge Removal on Graphs, J.
Comb. Math. Comb. Comp. 57 (2006), 75-82
[30] R. Gould, T. Luczak, J. Schmitt, Constructive upper bounds for cyclesaturated graphs of minimum size, Electron. J. Combin., 13(1):Research
Paper 29 (2006), 19
[31] J. Grossman, Graphs with Unique Ramsey Colorings, J. Graph Theory 7
(1983), 85-90
[32] A. Hajnal, A theorem on k-saturated graphs, Canad. J. Math. 17 (1965),
720-724
[33] G. Hajos, Über eine Art von Graphen, Internat. Math. Nachr., 47 (1957),
Problem 65
[34] D. Hanson, B. Toft, Edge-Colored Saturated Graphs, J. Graph Theory 11,
no 2 (1987), 191-196
[35] F. Harary, J. A. Kabell, and F. R. McMorris, Bipartite intersection graphs,
Comment. Math. Univ. Carolina 23(4) (1982), 739-745
[36] F. Havet, S. Thomassé, Oriented hamiltonian paths in tournaments: a proof
of Rosenfeld’s conjecture, J. Comb. Theory, Ser B 78 2000, 243-273
[37] P. Hell, Personal Correspondance
[38] P. Hell and J. Huang, Interval Bigraphs and Circular Arc Graphs, J. Graph
Theory 46 (2004), 313-327
[39] R. Isaacs, Infinite families of nontrivial trivalent graphs which are not Tait
colorable, Amer. Math. Monthly 82 (1975), 221-239
[40] R. Kalinowski, Z. Skupień, Large Isaacs’ Graphs are Maximally NonHamiltonian-Connected, Discrete Math. 82 (1990), 101-104
[41] L. Kaszonyi, Z. Tuza, Saturated graphs with minimal number of edges, J.
Graph Theory 10 (1986), 203-210
[42] K. Kuratowski, Sur le problme des courbes gauches en topologie, Fund.
Math. 15 (1930), 271-283
[43] C. G. Lekkerkerker and J.C. Boland, Representations of finite graphs by a
set of intervals on the real line, Fund. Math. 51 (1962), 45-64
104
[44] X. Lin, W. Jiang, C. Zhang, Y. Yang, On smallest maximally nonHamiltonian graphs, Ars Combin., 45 (1997), 263-270
[45] W. Mader, 3n − 5 edges do force a subdivision of K5 , Combinatorica 18
(1998), 569-595
[46] W. Mantel, Problem 28, solution by H. Gouwentak, W. Mantel, J. Teixeira
de Mattes, F. Schuh, W. A. Wythoff, Wiskundige Opgaven 10 (1907), 60-61
[47] H. Muller, Recognizing Interval Digraphs and Interval Bigraphs in Polynomial Time, Discrete Appl. Math. 78 (1997), 189-205
[48] L. Ollman, K2,2 saturated graphs with a minimal number of edges, Proceedings of the Third Southeastern Conference on Combinatorics, Graph
Theory, and Computing (1972), 367-392
[49] O. Pikhurko, Asymptotic evaluation of the sat-function for r-stars, Discrete
Math. 214(1-3) (2000), 275-278
[50] O. Pikhurko, Weakly saturated hypergraphs and exterior algebra, Combin.
Probab. Comput. 10(5) (2001), 435-451
[51] O. Pikhurko, Results and open problems on minmum saturated hypergraphs, Ars Combin. 72 (2004), 111-127
[52] F. P. Ramsey, On a problem of formal logic, Proc. London Math. Soc. 30
(1930), 264-286
[53] F. Roberts, Representations of Indifference Relations, PhD thesis, Department of Mathematics, Stanford University 1968
[54] C. Thomassen, Hamiltonian-Connected Tournaments, J. of Combin. Theory (1980), 142-163
[55] W. Trotter and J. Moore, Characterization problems for graphs, partially
ordered sets, lattices, and families of sets, Discrete Math. 16 (1976), 361381
[56] P. Turán, On the theory of graphs, Colloq. Math. 3 (1954), 19-30 MR
15,976b.
[57] Z. Tuza, C4 -saturated graphs of minimum size, Acta Univ. Carolin. Math.
Phys. 30(2) (1989), 161-167
105
[58] D. B. West, Introduction to Graph Theory, 2nd edition, Prentice Hall, Saddle River, NJ (2001)
106