RICE UNIVERSITY
New Sufficient Condition for Hamiltonian Paths
by
Landon Jennings
A THESIS SUBMITTED
IN PARTIAL FULFILLMENT OF THE
REQUIREMENTS FOR THE DEGREE
Doctor of Philosophy
Approved, Thesis Committee:
David Damanik, Associate Professor, Chair
Mathematics
Robin Forman, Professor and Dean of Undergraduates
Mathematics
Illya Hicks, Assistant Professor
Computational and Applied Mathematics
Houston, Texas
April 2008
ABSTRACT
New Sufficient Condition for Hamiltonian Paths
by
Landon Jennings
This paper proves a sufficient condition for the existence of Hamiltonian paths
in simple connected graphs. This condition was conjectured in 2006 by a computer
program named Graffiti.pc. Given examples will show that this new condition detects Hamiltonian paths that a theorem by Chvátal does not. A second condition
conjectured by Graffiti.pc is shown to satisfy Chvátal’s condition. Thus this second
conjecture, while true, does not improve on known results.
ACKNOWLEDGMENTS
I would like to thank my advisor, David Damanik, for his guidance and suggestions
through the writing phase of this work. His agreement to fill the role of advisor for
a student not working in his specific preferred area of mathematics will always be
appreciated.
Other faculty, such as Richard Stong, Tim Cochran, and Bob Hardt have also
provided valuable instruction and advice. I also want to take this opportunity to
thank a lot of mathematicians who helped me in various ways, in particular, Max
Warshaure, Terry McCabe, Cody Patterson, Ermelinda DeLaViña, Jackie Sack, Abel
Bourbois, Janet Schofield and Jon McCammond. A special thanks should be given
to Dr. Carol Hazlewood for being the first to introduce me to the wonderful world of
Graph Theory.
In addition, I would like to acknowledge the support of the Math Department
during my graduate study. Marie Magee and Maxine Turner have provided much
help with many details.
I would also like to thank all of my friends from the Math Department for providing
such a wonderful environment, not just for math, also for an easy, fulfilling life; in
particular Karoline, Jamie, Peter, Jon, Dave Casey, and Amanda.
Thank you to Rice University for being my home and teaching me so much the past
7 years. This work would not have been possible if not for the financial support of Rice
University, National Science Foundation Mathematics Leadership Institute Grant,
and National Science Foundation Vertical Integration of Research and Education
Grant.
Much thanks is also given to my family. They have loved, cared for, supported,
helped and yes even slightly discouraged me at times. All of which has brought me
to this point.
Contents
Abstract
ii
Acknowledgments
iii
Contents
iv
1 Preliminaries
1
1.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1
1.2
Basic Definitions and Notation . . . . . . . . . . . . . . . . . . . . . .
3
2 Graffiti and Sophie
5
2.1
Graffiti . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5
2.2
Graffiti.pc and Sophie . . . . . . . . . . . . . . . . . . . . . . . . . .
7
3 Lemmas and Previous Theorems
8
3.1
Forbidden Subgraph Conditions . . . . . . . . . . . . . . . . . . . . .
3.2
Degree Conditions
. . . . . . . . . . . . . . . . . . . . . . . . . . . .
10
3.3
Basic Lemmas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
15
4 Annihilation Number and Conjecture 188
8
20
4.1
Annihilation Number . . . . . . . . . . . . . . . . . . . . . . . . . . .
20
4.2
Conjecture 188 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
22
4.3
Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
29
5 Maximum Mode and Conjecture 206
32
5.1
Conjecture 206 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
32
5.2
Consequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
35
Bibliography
37
v
vi
List of Figures
1.1
Famous graphs with Hamiltonian paths and no Hamiltonian cycles . .
2
3.1
Forbidden Graphs in Theorem 3.1.1 . . . . . . . . . . . . . . . . . . .
9
3.2
Dirac and Ore do not apply here but Chvátal does. . . . . . . . . . .
12
4.1
Distinguishing Graphs for Conjecture 188 . . . . . . . . . . . . . . . .
30
4.2
Diagram for constructing examples of odd order . . . . . . . . . . . .
31
4.3
Diagram for constructing examples of even order . . . . . . . . . . . .
31
Chapter 1
Preliminaries
1.1
Introduction
If a graph is viewed as a representation of a finite set of objects and some set of
allowed connections between pairs of these objects, then a natural question is can
one follow these connections and visit each object in an efficient way? If an efficiency
requirement is that each object is visited exactly once, then we are asking if the graph
has a Hamiltonian path. If we are required to return to the object from which we
began, then we are asking if the graph has a Hamiltonian cycle. This problem was
considered as early as 1856 by Thomas Kirkman [28]. If we also assign costs to each
connection, then we have the full statement of the famous Traveling Salesman Problem
[31]. While the Hamiltonian cycle problem may be more famous, the author believes
the Hamiltonian path problem is at least as natural and possibly more important.
Both the Hamiltonian path and cycle problems are NP-complete [27]. So in
some sense they are mathematically equivalent. Chapter 3 of this paper shows that
any characterization for Hamiltonian cycles gives a characterization for Hamiltonian
paths. But in application, having a Hamiltonian cycle is more restrictive than having
a Hamiltonian path. Obviously every graph with a Hamiltonian cycle has a Hamilto-
1
2
(a) Petersen
(b) Herschel
(c) theta-7
Figure 1.1: Famous graphs with Hamiltonian paths and no Hamiltonian cycles
nian path, and the converse is not true. Figure 1.1 shows three examples of famous
graphs that have Hamiltonian paths but no Hamiltonian cycles.
The rest of this chapter provides a short graph theory background that will be
useful to the rest of this paper. If any terms used later are not made clear, the
author recommends [31] and [3] as good reference sources. Chapter 2 will discuss the
computer programs Graffiti and Graffiti.pc. There will be a very basic, nontechnical
description of how these programs work along with the evolution and accomplishments
made possible by each.
Chapter 3 will describe the major sufficient conditions for Hamiltonian paths that
can already be found in the literature. Easy corollaries will be proved here as well
as lemmas that will be useful in the subsequent chapters. Chapter 4 will contain the
proof of the main result in this paper which is the following theorem.
Graffiti.pc Conjecture (Conjecture 188). Let G be a connected graph. If σ(G) ≥
A(G) − 1 then G contains a Hamiltonian path.
Also there will be examples of graphs and discriptions showing that the new
condition presented holds for cases that previous theorems do not. Chapter 5 will
show that another conjecture made by Graffiti.pc is true but also weaker than a
theorem already described in Chapter 3. Here is the statement of that conjecture.
Graffiti.pc Conjecture (Conjecture 206). Let G be a connected graph. If m(G) ≥
2n(G) − 4 − 2σ(G), then G satisfies Chvátal’s condition.
3
1.2
Basic Definitions and Notation
This paper seeks to prove conjectures for sufficient conditions for Hamiltonian paths
in simple connected graphs. The considered conjectures are a subset of ones posed by
a computer program called Graffiti.pc. Since all simple connected graphs with three
or fewer vertices have Hamiltonian paths, all the graphs we will consider will have
order at least 4.
Definition. For a simple graph G, V (G) is the set of all vertices in G and E(G) is
the set of all edges in G.
Vertices of a graph will be represented by lower case english letters possibly with
subscripts like the following: v, u3 , or vi . Edges will be represented as the concatenation of the names of the two vertices it connects. For example vu, s1 v, and tn−3 v2
would all represent edges in a graph. It should be noted that since we are not dealing with directed graphs the order of the concatenation does not matter, therefore
vu = uv.
Definition. The degree of vertex v, deg(v), is equal to the number of distinct edges
to which is belongs.
Definition. The order of a graph G, n(G), is equal to the number of vertices present
in G.
When only one graph is being discussed n(G) may simply be written as n for ease
of notation. This shorthand is utilized for many graph invariants. In proofs, the use
of such shorthands will be made explicit. The following are definitions of basic graph
invariants.
Definition. An independent set of a graph G is a subset of the vertices S ⊂
V (G) such that there is no edge between any two elements of S. The independence
4
number of a graph G, α(G), is equal to the size of the largest independent set of
vertices in G.
One may think of independence number as a measure of how spread out the graph
is. If so, then the next invariant can be thought of measuring how linked the graph
is.
Definition. The connectivity of a graph G, κ(G), is equal to the minimum size
vertex set S such that G − S is disconnected or has only one vertex.
Graph invariants need not be just single numbers or numbers at all. Here is an
example of an important invariant that is a sequence of numbers.
Definition. Suppose G is a graph and let n = n(G). The degree sequence of the
graph G, (d1 , d2 , . . . , dn ), is a nondecreasing list of the degrees of the vertices in G.
For notational ease δ(G) = d1 , σ(G) = d2 , Σ(G) = dn−1 , and ∆(G) = dn .
Next we will define two operations on graphs that will be used later in this paper.
Definition. G ∨ H is the join of the graphs G and H. V (G ∨ H) = V (G) t V (G)
and E(G ∨ H) = E(G) t E(H) t {xy|x ∈ V (G), y ∈ V (H)}.
Definition. Suppose G is a graph and S ⊂ V (G). G without S, G \ S, is the graph
with vertex set V (G) \ S and edge set {uv|uv ∈ E(G) and u, v ∈
/ S}.
The next two definitions are for the graph substructures with which this paper is
most concerned.
Definition. A path P of order k in a graph G is a a sequence of vertices [pi ]k1 =
[p1 , p2 , . . . , pk ] such that each pi ∈ V (G), pi pi+1 ∈ E(G) and pi = pj if and only if
i = j.
Definition. A Hamiltonian path of a graph G is a path that contains every vertex
of G.
5
Other sources may refer to a Hamiltonian path simply as a spanning path or say
that the graph is traceable.
Chapter 2
Graffiti and Sophie
This section discusses the history of people and software that came together to eventually conjecture the statement of the main theorem of this paper. This will only
be a brief discussion of these topics. For more information and details please see
Ermelinda DeLaViña’s paper “Some History of the Development of Graffiti” [8].
2.1
Graffiti
In the mid-1980s Siemion Fajtlowicz wrote a computer program designed to make
graph theory conjectures. He called his program Graffiti. Fajtlowicz first submitted
conjectures from Graffiti in 1986. Since that time the power of this program has
grown because of increase in available computing power and in the sophistication of
its own code. From 1986 to 1995 Fajtlowicz has had five separate papers published
containing examples of open conjectures produced by Graffiti and information on the
program itself [17, 16, 18, 19, 20].
At first, Graffiti made conjectures of the from I + J ≤ K + L. The program would
consider these inequalities with the variable ranging over a set of over 60 invariants
along with constants 0 and 1 [17]. For each inequality Graffiti would test a database
of graphs searching for a counterexample. If no counterexample was found, then that
6
7
particular inequality was considered a conjecture. Further tests were then applied to
eliminate true but trivial conjectures. As computing hardware improved more graphs
were added to the database and more invariants were added to consider.
In the early 1990s, a significant change came to the code and application of Graffiti.
It was applied to the fields of geometry and number theory. The algebra of possible
expressions was enlarged. Graffiti was given more than just invariants and graphs.
Graffiti was given the ability to calculate graph theoretical properties. Now the
user could dictate that a certain property would be part of the hypothesis. These
properties could be, for example, bipartite or triangle-free. Then a heuristic was
written to have Graffiti produce new properties that might be interesting on its own.
One such example is “the class of all graphs in which the smallest eigenvalue has
multiplicity 1” [20].
The author finds the next evolution of Graffiti especially interesting. A new
heuristic was written called Dalmatian. The addition of Dalmatian allowed Graffiti to
take into account the informativeness of its conjectures. Each time a new conjecture
was produced Graffiti would test it against the previous conjectures. If the graph
database did not contain a graph to which the new conjecture applied and the previous
ones did not, then the new conjecture was considered uninformative and therefore
discarded. Now if a conjecture produced is true, then there is a at least one example
graph to which it applies.
One can view many of the conjectures produced by Graffiti at [15]. Through the
years, Graffiti conjectures have inspired many papers and theses. Some of the more
notable authors who have proved Graffiti conjectures are the following [8]: Alon,
Bollobás, Chung, Erdös, Kleitman, Lovász, Pach, Seymour, Shearer and Spencer. A
list, [6], is kept to document how Graffiti has contributed to mathematics.
8
2.2
Graffiti.pc and Sophie
In 2001 DeLaViña developed a program called Graffiti.pc [8]. One goal of this was
to have a program similar to Graffiti on the PC platform. There was also a purpose
to use this new program as a teaching tool in undergraduate research. The first
application of Graffiti.pc used a heuristic very similar to Dalmatian. In fact, even the
same name was used. Similar to Graffiti a list is maintained at [7] of the conjectures
produced by Graffiti.pc. A more complete description of the program can be found
at [11].
In 2006, DeLaViña gave a talk at the annual CombinaTexas conference [9]. This
talk presented conjectures made by the new heuristic Sophie, developed by DeLaViña
and Bill Waller, under Graffiti.pc [10]. Previously Graffiti was used to conjecture
mathematical expressions necessary for a certain class of graphs. Sophie is designed
to find expressions that are sufficient for a graph property chosen by the user.
The first testbed for Sophie was graphs containing Hamiltonian paths. Graffiti.pc
produced 34 distinct conjectures of sufficient conditions for Hamiltonian paths. A few
of these have been resolved as being either true or false, but most are still open. The
main theorem of this paper is taken from these conjectures. It is labeled as Conjecture
188 in [7]. The rest of this paper is concerned with the proof of this conjecture.
Chapter 3
Lemmas and Previous Theorems
More work has been done in the past on sufficient conditions for Hamiltonian (or
spanning) cycles than on Hamiltonian paths.
k
Definition. C = [v1 , v2 , . . . , vk ] = [vi ]i=1 is a cycle of order k in a graph G if
[v1 , v2 , . . . , vk ] is a path in G and v1 vk ∈ E(G). If k = n(G) then C is called a
Hamiltonian cycle.
It is clear that Hamiltonian paths and Hamiltonian cycles are closely related. From
the above definition it is obvious that a graph with a Hamiltonian cycle also contains
a Hamiltonian path. Therefore any sufficient condition for Hamiltonian cycles is also
sufficient for Hamiltonian paths.
3.1
Forbidden Subgraph Conditions
Definition. A graph G is said to be {F1 , F2 , . . . , Fk }-free if G contains no induced
subgraph isomorphic to any Fi , 1 ≤ i ≤ k.
There are many theorems in graph theory that use this idea of forbidden subgraphs. This is especially true for sufficient conditions. Here is one such sufficient
condition for Hamiltonian cycles.
9
10
(a) K1,3 or claw
(b) N1,1,1 or net
Figure 3.1: Forbidden Graphs in Theorem 3.1.1
Theorem 3.1.1 (Duffus, Gould and Jacobson [13]). If G is a 2-connected, {K1,3 , N1,1,1 }free graph, then G has a Hamiltonian cycle.
K1,3 and N1,1,1 are shown in Figure 3.1 and known as a claw and net respectively.
The condition of being 2-connected is not unreasonable here because 2-connected
is also a necessary condition for a Hamiltonian cycle. Of course 1-connected, or simply
connected, is a necessary condition for Hamiltonian paths. Therefore the author sees
connectedness as a reasonable condition to require, if it allows for results beyond what
is already known. In the next section, conditions for Hamiltonian paths will be given
that do not require connectedness to be checked separately, but the main theorem of
this paper does have that requirement.
There are many other forbidden pairs that produce Hamiltonian cycles [1]. You
can find examples of such theorems in [25]. Any such pair requires K1,3 to be one of the
forbidden graphs and the graph to be 2-connected [24]. If you allow three forbidden
graphs, then K1,3 is no longer required (but of course 2-connected still is)[23]. [24]
also shows that the only single forbidden subgraph that implies Hamiltonian cycles
is P3 . Of course if a graph contains no P3 as an induced subgraph and has at least 3
vertices, then it must be a complete graph.
Theorem 3.1.1 stands out among the other forbidden pair of graphs results in that
if 2-connected is relaxed to just connected we get a corollary for Hamiltonian paths.
Corollary 3.1.2. If G is a connected, {K1,3 , N1,1,1 }-free graph, then G has a Hamil-
11
tonian path.
Example graphs will be given that fail this condition but fulfill the condition of
the main theorem.
3.2
Degree Conditions
One of the strongest such theorems is the following:
Theorem 3.2.1 (Chvátal [4]). Let d1 ≤ d2 ≤ · · · ≤ dn be the degrees of G. If
dk ≤ k ≤
n
⇒ dn−k ≥ n − k
2
then G has a Hamiltonian cycle.
The strength of this theorem is seen by showing that the famous older thereoms
are strictly weaker.
Corollary 3.2.2 (Dirac [12]). Let G be a simple graph of order n. If δ(G) ≥
n
2
then
G has a Hamiltonian cycle.
Proof. Let d1 ≤ d2 ≤ · · · ≤ dn be the degrees of G. Since δ(G) ≥ n2 , then di ≥
n
2
for
all i. Thus
dk ≤ k ≤
n
⇒ dn−k ≥ n − k
2
is vacuously satisfied. Therefore G satisfies the hypothesis of Theorem 3.2.1.
Corollary 3.2.3 (Ore [29]). Let G be a simple graph of order n. If deg(u)+deg(v) ≥ n
for all distinct, nonadjacent u, v ∈ V (G), then G has a Hamiltonian cycle.
Proof. Let deg(v1 ) ≤ deg(v2 ) ≤ · · · ≤ deg(vn ) be the degrees of G. Suppose G does
not satisfy the hypothesis of Theorem 3.2.1.
12
This means there is some integer k such that deg(vk ) ≤ k ≤
n
2
and deg(vn−k ) <
n − k.
deg(vk ) + deg(vn−k ) < n.
deg(vk ) + deg(vi ) < n for i ≤ n − k.
So vk must be adjacent to all other vertices with index less than or equal to n − k
by the hypothesis. So n − k − 1 ≤ deg(vk ) ≤ deg(vn−k ) ≤ n − k − 1. This means
deg(vk ) = deg(vn−k ) = n−k−1. Since n−k−1 = deg(vn−k ) = deg(vk ) ≤ k then for all
i, j ∈ {1, 2, . . . , n−k}, deg(vi )+deg(vj ) ≤ k+(n−k−1) < n. vi ∈ V (G) for i ≤ n−k
form a complete graph G. Since deg(vn−k ) = n − k − 1 this complete sub graph is
also a component of G.
Since vn is not adjacent to any of the vertices of index less than n − k, then
k + 1 ≤ deg(vn ). Otherwise deg(vn ) + deg(vi ) < n. So vn can only be adjacent to
vertices with index greater than n − k. Thus k + 1 ≤ deg(vn ) < k.
This is a contradiction. Therefore G satisfies the hypothesis of Theorem 3.2.1.
Figure 3.2 shows an example of a graph that satisfies the hypothesis of Theorem
3.2.1 but not that of Corollaries 3.2.2 or 3.2.3. This graph is of order seven and has
two nonadjacent vertices of degree equal to three. This example along with the proofs
given here for the corollaries show that Theorem 3.2.1 is strictly stronger. In fact, if
some graph, G, fails to meet this condition, then there is a graph, H, on the same
number of vertices such that H does not have a Hamiltonian cycle and the degree
sequence of H majorizes the degree sequence of G. This means that every entry in
the degree sequence of H is greater than or equal to the corresponding entry in the
degree sequence of G.
The graph being simple is the only requirement beyond the degree sequence that
Chvátal’s cycle condition makes. This condition is strong enough to not even require
connectedness to be checked seperately. With other requirements for structure other
13
Figure 3.2: Dirac and Ore do not apply here but Chvátal does.
statements can be made.
Theorem 3.2.4 (Jackson [26]). Let G be a simple, regular, 2-connected graph of
order n. If the degree of G is at least n3 , then G has a Hamiltonian cycle.
And here is a theorem that used other graph invariants besides the degrees.
Theorem 3.2.5 (Chvátal and Erdös [5]). If κ(G) ≥ α(G) then G has a Hamiltonian
cycle.
Using the structure of Hamiltonian cycles we have the following complete characterization of Hamiltonian paths.
Theorem 3.2.6. Let G be a graph of order n and v a vertex not in G. G ∨ v contains
a Hamiltonian cycle if and only if G contains a Hamiltonian path.
Proof. (Sufficiency) Let C be a Hamiltonian cycle in G ∨ v. Then C can be written
as [v, u1 , u2 , . . . , un ]. By the definition of cycle [v, u1 , u2 , . . . , un ] is a path in G ∨ v.
Therefore P = [u1 , u2 , . . . , un ] is a path in G. Since the order of P is equal to the
order of G, P is a Hamiltonian path of G.
(Necessity) Let P = [u1 , u2 , . . . , un ] be a Hamiltonian path in G. Since vu1 , vun ∈
E(G ∨ v), then C = [v, u1 , . . . , un ] is a Hamiltonian cycle in G ∨ v.
14
This simple result is found in most standard graph theory books like Introduction
to Graph Theory [31] by Douglas West. In application, this theorem only transfers
a person from one NP-complete problem to another. But while any sufficient condition for Hamiltonian cycles works also for Hamiltonian paths, this theorem allows
a condition that is sufficient for Hamiltonian cycles to possibly be relaxed to a less
restrictive condition for Hamiltonian paths.
First we will consider Dirac’s theorem which is Corollary 3.2.2.
Corollary 3.2.7 (Dirac [12]). Let G be a simple graph of order n. If δ(G) ≥
n−1
2
then G has a Hamiltonian Path.
Proof. Let G0 = G ∨ v. δ(G0 ) = δ(G) + 1 ≥
n+1
2
and the order of G0 is n + 1. So by
Corollary 3.2.2 G0 has a Hamiltonian cycle. Thus G has a Hamiltonian path.
Ore’s condition can be extended in a similar way to give the following statement.
Corollary 3.2.8. Let G be a simple graph of order n. If deg(u) + deg(v) ≥ n − 1 for
all distinct, nonadjacent u, v ∈ V (G), then G has a Hamiltonian path.
Here we can see Theorem 3.2.6 applied to Theorem 3.2.5.
Corollary 3.2.9 (Chvátal and Erdös [5]). If κ(G) + 1 ≥ α(G) then G has a Hamiltonian path.
Proof. Let G0 = G ∨ v. α(G0 ) = α(G) because any independent set of G is an
independent set of G0 and v is connected to all other vertices. Since v is connected
to all other vertices, any disconnecting set must include v. Once v is deleted, κ(G)
vertices must be deleted to disconnect the graph. Thus κ(G0 ) = κ(G) + 1. Therefore
κ(G0 ) ≥ α(G0 ) and by Theorem 3.2.5 G0 contains a Hamiltonian cycle. Theorem 3.2.6
implies G has a Hamiltonian path.
An extensive collection of sufficient conditions for Hamiltonian cycles can be found
in Claude Berge’s book Graphs and Hypergraphs [2]. The author does not know if
15
these theorems can be or have been extended to conditions for Hamiltonian paths
using Theorem 3.2.6.
If we apply Theorem 3.2.6 to Chvátal’s cycle condition we get the following powerful statement for Hamiltonian paths.
Corollary 3.2.10 (Chvátal’s Condition [4]). Let d1 ≤ d2 ≤ · · · ≤ dn be the degrees
of G. If
dk ≤ k − 1 ≤
n
− 1 ⇒ dn+1−k ≥ n − k
2
then G has a Hamiltonian path.
Note that this condition still does not require connectedness to be checked separately. If we restrict ourselves to the class of connected graphs other conditions can
be made that apply to graphs which Chvátal’s condition does not. Also this condition
inherits the notion of being the “best possible.” If a degree sequence fails Chvátal’s
condition then it can be majorized by a degree sequence of a graph that does not
have a Hamiltonian path [4].
For now we can use this corollary to prove a condition on just the second smallest
degree of the graph that will assure a Hamiltonian path. This condition will be used
in a later proof of a lemma that guarantees long paths even in graphs that do not
have a Hamiltonian path.
Corollary 3.2.11. Let G be a connected graph. If σ(G) ≥
n−1
then G satisfies
2
Chvátal’s condition and therefore has a Hamiltonian path.
Proof. Let d1 ≤ d2 ≤ · · · ≤ dn be the degrees of G. Since G is connected, 1 ≤ dk >
k − 1 where k = 1. For 2 ≤ k ≤ n2 ,
dk ≥
n−1
> k − 1.
2
Therefore G satisfies Chvátal’s condition.
16
This corollary is a significant improvement over the similar statement made about
δ(G). Once we required that G be connected the actual value of δ(G) can be ignored.
3.3
Basic Lemmas
Naturally the proofs to come will be concerned with assuring long paths in connected
graphs. So it will be useful to have the following lemma about maximum paths in
general.
Lemma 3.3.1. Let P = [vi ]ki=1 be a path of odd order with k ≥ 3 in graph G and u
k−1
be a vertex in G not on P . Suppose u has
neighbors on P , then u, v1 , v3 , . . . vk
2
forms an independent set or there exists a path of order k + 1 containing all the
vertices in P and u with one end point still v1 and the other is either u or vk .
Proof. Suppose that there is no path in G of order k + 1 that contains all the vertices
of P and u with one end point still v1 and the other is either u or vk . Obviously
u cannot be adjacent to v1 or vk . If u is adjacent to vi then u can not be adjacent
k−1
neighbors on P , then those neighbors must be exactly
to vi+1 . Since u has
2
{v2 , v4 , . . . , vk−1 }.
Assume vi vj ∈ E(G) for some distinct i, j ∈ {1, 3, . . . , k}. Without loss of
generality let i < j. If j = k then [v1 , v2 , . . . , vi , vk , vk−1 , . . . vi+1 , u] is a path in G.
If j 6= k then [v1 , v2 , . . . , vi , vj , vj−1 , . . . vi+1 , u, vj+1 , vj+2 , . . . , vk ] is a path in G. In
either case we have a path of order k + 1 containing the vertices of P and u. This is
a contradiction.
Therefore u, v1 , v3 , . . . vk forms an independent set.
In application, if P is known to be a maximum path in G, then this lemma
guarantees an independent set of a certain size. Next we will extend the ideas in the
17
proof of Dirac’s theorem for Hamiltonian cycles in order to get lower bounds on the
order of maximum paths in G.
Lemma 3.3.2. A connected graph G has a path of order at least δ(G) + σ(G) + 1, or
a Hamiltonian path.
Proof. Assume G does not have a Hamiltonian path. Let P = [vi ]k1 be a path in G of
maximum order with deg(v1 ) ≤ deg(vk ).
Let A = {i|v1 vi+1 ∈ E(G)} and B = {i|vk vi ∈ E(G)}. Since P is a maximum
path, all neighbors of v1 and vk are members of P . Since G is connected and has no
Hamiltonian path, if v1 vi+1 , vk vi ∈ E(G) then the vertices of P would form a cycle
and a longer path could be found. However, P was a longest path, so A ∩ B is empty.
A and B are both subsets of {1, 2, . . . , k − 1}.
0 = |A ∩ B| = |A| + |B| − |A ∪ B|.
0 ≥ δ(G) + σ(G) − (k − 1).
k ≥ δ(G) + σ(G) + 1.
An immediate corollary can be found requiring P to contain any prescribed vertex
of G. If v is the vertex we want to include, then all we must do is let P in the proof
be the maximum path in G containing v.
Corollary 3.3.3. Let G be a connected graph and vertex v ∈ V (G). Either G has a
Hamiltonian path, or path of order at least δ(G) + σ(G) + 1 containing v.
With slightly more work a much better fact can be proved that is still in the
flavor of Dirac’s result for Hamiltonian cycles. To do this we will use Corollary
3.2.11. This is a useful lemma that stretches the ideas of Dirac’s theorem and proof
18
for Hamiltonian cycles. The conclusion here gives a lower bound for the longest path
in a graph. This lower bound was already known and is proved by Erdös and Gallai
by a different method [14]. However, their statement did not include the statement
found here about the vertex of least degree. Because this proof is extended from
Dirac’s original ideas, the lemma is named in his honor.
Lemma 3.3.4 (Dirac for Paths). Let G be a connected graph. Then G contains a
path of order at least 2σ(G) + 1 or it has a Hamiltonian path. Moreover, if δ(G) ≥ 2
then there is a path in G of order at least 2σ(G) + 1 with a vertex of minimum degree
on its interior.
Proof. Let n = |V (G)|. Assume G does not have a Hamiltonian path (this implies
that n ≥ 4). Let σ = σ(G).
Case 1: δ(G) = 1.
Let v ∈ V (G) such that deg(v) = 1. Let G0 = G \ {v}. Since deg(v) = 1 and n ≥ 4,
G0 is a connected graph. We know from the Corollary 3.3.3, that G0 contains a path
of order at least δ(G0 ) + σ(G0 ) + 1 or G0 has a Hamiltonian path.
Claim 3.3.4.1. If G0 has a Hamiltonian path, then the theorem is satisfied.
Therefore we will suppose that G0 does not have a Hamiltonian path. Let u be
the neighbor of v in G. degG (u) ≥ σ. If degG (u) > σ then, δ(G0 ) = σ and σ(G0 ) ≥ σ.
This would tell us that there is a path in G0 , and hence a path in G, of length at least
2σ + 1. So we assume that degG (u) = σ. This forces δ(G0 ) = σ − 1 and σ(G0 ) ≥ σ.
Let [v1 , v2 , . . . , vk ] be a path of maximum order in G0 with deg(v1 ) ≤ deg(vk ).
Then k ≥ deg(v1 ) + deg(vk ) + 1. If v1 6= u then k ≥ 2σ + 1. So we will assume that
v1 = u. This means k ≥ 2σ. Since [u, v2 , . . . , vk ] is a path in G0 , then [v, u, v2 , . . . , vk ]
is a path of order at least 2σ + 1 in G.
Case 2: δ(G) ≥ 2.
19
Let P = [vi ]k1 be a maximum path in G containing a vertex of minimum degree labeled
such that deg(v1 ) ≤ deg(vk ).
Claim 3.3.4.2. P can be arranged so that σ ≤ deg(v1 ).
Let A = {i|v1 vi+1 ∈ E(G)} and B = {i|vk vi ∈ E(G)}. Since P is a maximum
path, all neighbors of v1 and vk are members of P . Since G is connected and has no
Hamiltonian path, if v1 vi+1 , vk vi ∈ E(G) then the vertices of P would form a cycle
and a longer path could be found. However, P was a longest path, so A ∩ B is empty.
A and B are both subsets of {1, 2, . . . , k − 1}.
0 = |A ∩ B| = |A| + |B| − |A ∪ B|.
0 ≥ σ(G) + σ(G) − (k − 1).
k ≥ 2σ(G) + 1.
Proof of Claim 3.3.4.1. Suppose G0 contains a Hamiltonian path. Since G does not
contain a Hamiltonian path, the longest path in G is of order n − 1. If this violates
the theorem we get the following equations:
n ≤ 2σ + 1
n−1
≤ σ.
2
By Theorem 4 this implies G has a Hamiltonian path which is a contradiction to the
assumptions at the beginning of the proof.
Proof of Claim 3.3.4.2. If deg(v1 ) ≥ σ, then we are done. So we shall assume that
20
deg(v1 ) = δ. Since v1 is the end of the maximum path P , all of its neighbors must be
vertices in P as well. One such neighbor is of course v2 , but since deg(v1 ) ≥ 2, there
is a vi ∈ P with i 6= 2 such that v1 vi ∈ E(G). This allows for the creation of a new
path, P 0 = [vi−1 , vi−2 , . . . , v1 , vi , vi+1 , . . . , vk ]. Then endpoints of P 0 are vi−1 and vk
which both have degrees ≥ σ. P 0 fits all the requirements for P in the proof so this
is the maximum path we shall use after relabeling.
Chapter 4
Annihilation Number and
Conjecture 188
Lemma 3.3.4 can only achieve, by itself, what was done in Corollary 3.2.11 for Hamiltonian paths. But we are going to use this lemma to find long paths which may not
necessarily be Hamiltonian but still useful.
Conjecture 188 uses a relatively new graphical invariant called the annihilation
number. This invariant was introduced by Ryan Pepper in [30] while exploring independence numbers [21]. Fortunately, Pepper was working close to the developer of
Graffiti.pc and this new invariant was added to those that had already been used.
4.1
Annihilation Number
While the idea for annihilation number originated with Pepper, the definition given
here is due to Siemion Fajtlowicz [22].
Definition. Let d1 ≤ d2 ≤ · · · ≤ dn be the degree sequence of a graph G. The
k
X
annihilation number of G, A(G) is the largest integer k such that
di ≤ |E(G)|.
i=1
From the definition one can see that this is a relatively simple invariant to com21
22
pute even if its uses are not immediately apparent. First the annihilation number
only depends on the degrees of the graph. Independence number and connectivity
both require the structure of the graph beyond the degree sequence for computation.
Second, the computer time required to calculate A(G) is linear in the number of edges
in G. Here are some easily verifiable facts about A(G).
Fact. α(G) ≤ A(G).
Fact.
n(G) − 1
≤ A(G).
2
Fact. If
k
X
i=1
di ≤
n
X
di , then k ≤ A(G).
i=k+1
Annihilation number being small can produce more structure in the degree sequence and therefore the graph. The general idea is that the smaller the annihilation
number, the more regular the graph becomes. Here are two easy lemmas concerning
the minimum values of annihilation number.
Lemma 4.1.1. Let G be a simple graph of order n with degree sequence d1 ≤ d2 ≤
· · · ≤ dn . If d1 + d2 ≤ dn , then A(G) ≥
n+1
.
2
Proof.
d1 + d2 ≤ dn ,
b n+2
c
2
X
di ≤
b n+2
c
2
X
di ≤
b n+2
c
2
i=1
Therefore A(G) ≥ b n+2
c≥
2
n+1
.
2
n−1
X
di + dn ,
c
b n+4
2
i=3
X
di ,
b n+4
c
2
i=3
d1 + d2 +
n−1
X
di ≤
n
X
b n+4
c
2
di .
23
Lemma 4.1.2. If G is a graph of odd order, n(G) = 2k − 1, and A(G) =
n−1
2
= k − 1,
then
dk >
k−2
X
(dn−i − di+1 ).
i=0
Proof.
k−1
A(G) =
k−2
X
X
n−1
=k−1⇒
di+1 >
dn−i
2
i=0
i=0
⇒ dk >
⇒ dk >
k−2
X
i=0
k−2
X
dn−i −
k−2
X
di+1
i=0
(dn−i − di+1 )
i=0
The author found Lemma 4.1.2 particularly enlightening and useful during exploration. One of the main reasons for this is that if ai = (dn−i − di+1 ) is viewed as a
sequence, then one can use the fact that it is non-decreasing.
4.2
Conjecture 188
Now we come to the statement and proof of the main theorem of this paper which
was conjectured by Graffiti.pc [7].
Theorem 4.2.1 (Conjecture 188). Let G be a connected graph. If σ(G) ≥ A(G) − 1
then G contains a Hamiltonian path.
Proof. Let n = n(G), σ = σ(G) and δ = δ(G). Assume G does not contain a
Hamiltonian path.
Since G does not have a Hamiltonian path, Lemma 3.3.4 implies
n−3
n−2
≤ A(G) − 1 ≤ σ ≤
.
2
2
24
Case 1: n(G) is odd.
Since A(G) and σ must be integers, we know A(G) =
n−1
2
and σ(G) =
n−3
.
2
Claim 4.2.1.1. δ ≥ 2
Claim 4.2.1.1 together with Lemma 3.3.4 imply, there exists a path in G of order
at least n − 2 that contains a vertex of smallest degree on its interior. Let P = [vi ]ki=1
be a maximum path containing a vertex of degree δ on its interior. Since G does not
have a Hamiltonian path, the order of P is either n − 2 or n − 1.
Claim 4.2.1.2. P has order n − 1.
Let u ∈ V (G) be the vertex not contained in P . Since a vertex of degree δ is
on the interior of P , deg(u) ≥ σ =
n−3
.
2
Since P is a maximum path, u cannot be
adjacent to v1 , vn−1 nor both of any pair vi and vi−1 .
Let S = {i|uvi+1 ∈ E(G)} ∪ {n − 1}. For all i ∈ S vi u 6∈ E(G). |S| = deg(u) + 1 ≥
n−1
.
2
Claim 4.2.1.3. If i, j ∈ S, then vi vj ∈
/ E(G).
Therefore {vi }i∈S ∪ {u} forms an independent set of size at least
n+1
.
2
Thus
n+1
n−1
≤ α(G) ≤ A(G) =
.
2
2
This is a contradiction.
Case 2: n(G) is even.
Since n is even, A(G) ≥
n
.
2
Thus σ ≥ A(G) − 1 ≥
n−2
.
2
By Lemma 3.3.4, there is
a path in G of order at least 2σ + 1. Since G does not have a Hamiltonian path,
A(G) =
n
2
and σ =
n−2
.
2
Thus there is a path of order n − 1.
Claim 4.2.1.4. There is a path P = [v1 , v2 , . . . vn−1 ] of order n − 1 that contains a
vertex of minimum degree.
25
Let u be the vertex in G that is not on the path P. Since there is a vertex of
minimum degree on the interior of P, then deg(u) ≥ σ =
n−2
.
2
G does not contain a
Hamiltonian path, which means u cannot be adjacent to v1 , vn−1 , nor any consecutive
vertices on P . Thus deg(u) =
n−2
.
2
n − 1 is odd so by Lemma 3.3.1, either there
is a path of order n or {u, v1 , v3 , v5 , . . . , vn−1 } is an independent set. There is no
n+2
. This introduces some
Hamiltonian path so we have an independent set of size
2
new inequalities.
n+2
≤ α(G).
2
α(G) ≤ A(G).
n
.
2
n+2
n
≤
.
2
2
A(G) =
The last line is an obvious contradiction.
Both cases end in contradiction. Therefore the assumption that G does not have
a Hamiltonian path is false.
Proof of Claim 4.2.1.1. Assume δ = 1. Let n = 2k + 1.
Pn
i=2
di = 2|V (G)| − 1 is the
sum of an even number of summands with an odd total. Therefore not all di are equal
for i ≥ 2. Since degree sequences are nondecreasing, σ + 1 ≤ dn and d2+i ≤ dk+1+i for
P
P2k
Pk+1
Pn
1 ≤ i ≤ k − 1. Thus k+1
d
≤
d
which
implies
d
≤
i
i
i
i=3
i=k+2
i=1
i=k+2 di . This
means that
n−1
n+1
= A(G) ≥
.
2
2
This is a contradiction so δ ≥ 2.
Proof of Claim 4.2.1.2. Since the order of P is either n − 1 or n − 2, assume for the
sake of contradiction that the order of P is n − 2. Let the two vertices not contained
in P be labeled s and t. Of course either st is a vertex in G or it is not.
26
Case 1: st 6∈ E(G).
Since P is a maximum path in G with a vertex of minimum degree on its interior,
by Lemma 3.3.1 {s, v1 , v3 , . . . , vn−2 } and {t, v1 , v3 , . . . , vn−2 } form independent sets.
Since st ∈
/ E(G), then S = {s, t, v1 , v3 , . . . , vn−2 } in an independent set in G. But
|S| =
n−1
n−1
+ 2 ≤ α(G) ≤ A(G) =
2
2
This is a contradiction. So the order of P must be n − 1.
Case 2: st ∈ E(G)
Since P is a maximum path in G with a vertex of minimum degree on its interior,
s can not be adjacent to either v1 nor vn−2 . Also if s is adjacent to some vi then
svi+1 ∈
/ E(G). If sv2 ∈ E(G) then [t, s, v2 , v3 , . . . , vn−2 ] would be a path in G. But
this would contradict the the fact that P is a maximum. So s is not adjacent to
v2 nor similarly vn−3 . Since deg(s) ≥ σ and st ∈ E(G) then s must be adjacent to
n−3
at least
vertices of P . Therefore the vertices adjacent to s must be exactly
2
S = {t, v3 , v5 , . . . , vn−4 }. By the symmetry of s and t, the vertices adjacent to t must
be exactly T = {s, v3 , v5 , . . . , vn−4 }.
Small cases for this proof will require special attention. The following is a calculation of a lower bound on n.
2 ≤ δ ≤ σ.
2 ≤
n−3
.
2
7 ≤ n.
Since n ≥ 7 and odd, we will use the subcases of n = 7, n = 9, and n ≥ 11. Let
(d1 , d2 , . . . , dn ) be the degree sequence for G.
Subcase 1: n = 7
27
Recall 2 ≤ δ ≤ σ =
n−3
2
= 2. So the first two terms in the degree sequence are equal to
2. Recall also that A(G) =
n−1
2
= 3. We know that v3 is adjacent to v2 , v4 , s and, t.
Thus the largest term in the degree sequence dn ≥ deg(v3 ) ≥ 4.
2+2 ≤ 4
d1 + d2 ≤ dn
d3 + d4 ≤ d5 + d6
4
7
X
X
di ≤
di .
i=1
i=5
Therefore A(G) ≥ 4. This is a contradiction because A(G) = 3.
Subcase 2: n = 9
Recall σ =
n−3
2
= 3. So the first two terms in the degree sequence are less than or
equal to 3. Recall also that A(G) =
n−1
2
= 4. s is adjacent to exactly t, v3 , and, v5 ,
and t is adjacent to exactly s, v3 , and, v5 . Thus deg(s) = deg(t) = 3, deg(v3 ) ≥ 4,
and deg(v3 ) ≥ 4. If deg(v4 ) ≥ 3, then [v1 , v2 , v3 , s, t, v5 , v6 , v7 ] is a path of length n − 1
with a vertex of minimum degree on the interior which would contradict the fact that
P is a maximum. Therefore deg(v4 ) = 2. So the degree sequence begins with 2, 3, 3
and ends with two numbers each at least 4.
2+3+3 ≤ 4+4
d1 + d2 + d3 ≤ d8 + d9
d4 + d5 ≤ d6 + d7
5
9
X
X
di ≤
di .
i=1
i=6
Therefore A(G) ≥ 5. This is a contradiction because A(G) = 4.
Subcase 3: n ≥ 11
28
Consider the vertices v4 and v6 . Let Q = [v1 , v2 , v3 , s, t, v5 , v6 , . . . vn−2 ] and R =
[v1 , v2 , . . . , v5 , s, t, v7 , v8 , . . . , vn−2 ]. Each of these paths is of order n − 1. Q misses the
vertex v4 and R misses v6 . If v4 is of minimum degree then R yields a contradiction.
If v4 is not of minimum degree then Q yields a contradiction. Since at least one of
these statements is true, we have a contradiction.
So in all cases we have a contradiction. Therefore the order of P must be n − 1.
Proof of Claim 4.2.1.3. Assume vi vj ∈ E(G). Since i ∈ S and i < j, uvi+1 ∈ E(G).
Since j ∈ S, either uvj+1 ∈ E(G) or j = n − 1.
Case 1: uvj+1 ∈ E(G).
[v1 , v2 , . . . , vi , vj , vj − 1, . . . , vi+1 , u, vj+1 , vj+2 , . . . , vn−1 ] is a path in G. Thus G has a
Hamiltonian path.
Case 2: j = n − 1. [v1 , v2 , . . . , vi , vn−1 , vn−2 , . . . , vi+1 , u] is a path in G. Thus G has
a Hamiltonian path.
Both cases conclude that G has a Hamiltonian path. This is a contradiction to
the assumption about G made in the proof. So vi vj ∈
/ E(G).
Proof of Claim 4.2.1.4. Assume there is no path in G of order n − 1 that contains a
vertex of minimum degree. Lemma 3.3.1 therefore implies that δ = 1. Let u be the
vertex in G of degree 1.
Lemma 3.3.4 assures a path in G of order n − 1. Label this path P = [vi ]n−1
i=1 .
Since u cannot be a vertex of P , let k such that vk u ∈ E(G). There does not exist
an i such that vi+1 v1 and vi vn−1 ∈ E(G), because G \ {u} would have a Hamiltonian
cycle and therefore G would have a Hamiltonian path. There also cannot be an i such
that v1 vi+1 and vn−1 vi−1 . If there were, then i = k would give rise to a Hamiltonian
path and i 6= k would give rise to a path in G of order n − 1 that contains u.
29
Suppose there exists i such that 3 ≤ i ≤ n − 4 and neither v1 nor vn−1 is adjacent
to either of vi or vi+1 . Let A = {j|v1 vj+1 ∈ E(G)} and B = {j|vn−1 vj ∈ E(G)}.
Since G is connected and has no Hamiltonian path, if v1 vj+1 , vn−1 vj ∈ E(G) then
the vertices of P would form a cycle and a longer path could be found. However, P
is a longest path, so A ∩ B is empty. Since v1 is not adjacent to vi+1 , then i ∈
/ A.
Since vn−1 is not adjacent to vi , then i ∈
/ B. Since G has no loops and there is no
vertex with index equal to n, then n − 1 ∈
/ A ∪ B. So A and B are both subsets of
{1, 2, . . . , i − 1, i + 1, i + 2, . . . , n − 2}.
0 = |A ∩ B| = |A| + |B| − |A ∪ B|.
0 ≥ σ + σ − (n − 3).
0 ≥ (n − 2) − (n − 3).
0≥1
This contradiction shows that no such i can exist. Or put another way, no two
consecutive vertices of P can both be independent of v1 and vn−1
Let m be the largest integer such that v1 vm ∈ E(G). Since all of the neighbors of
v1 are on P , m ≥ n2 . Since vn−1 must not be adjacent to vm−1 nor vm−2 , then v1 must
be adjacent to one of vm−1 or vm−2 . This argument can then be repeated. Ultimately
this will imply that vn−1 vi ∈
/ E(G) for any i < m. Thus all of the neighbors of vn−1
have indices greater than or equal to m. So we know m ≤ n2 , and therefore m = n2 .
v1 is then connected to all other vertices of indices less than or equal to n2 . Similarly, vn−1 is connected to all other vertices of indices greater than or equal to n2 .
Suppose i <
n
2
and j >
n
2
such that vi vj ∈ E(G). Then we have the following
30
Hamiltonian cycle for G \ {u}:
[v1 , v2 , . . . , vi , vj , vj+1 , . . . vn−1 , vj−1 , vj−2 , . . . , vi+1 , v1 ].
This gives rise to a Hamiltonian path in G. So no such i and j can exist. Thus there
are no edges connecting the vertices of index less than m to the vertices of index
greater than m.
All vertices of P have degree at least
n−2
.
2
This means that all vertices of index
less than m must be adjacent to vm . Similarly all vertices of index greater than m
must also be adjacent to vm . The degree of vm must then be greater than or equal to
n − 2.
1+
n−2
≤ n−2
2
d1 + d2 ≤ dn
n
+1
2
X
di ≤
n
i=1
This implies that A(G) =
n+2
.
2
di
+2
i= n
2
i=3
+1
2
X
n−1
X
di ≤
n
X
di .
i= n
+2
2
But A(G) = n2 . This contradiction confirms that
there is a path in G containing a vertex of minimum degree.
4.3
Examples
This section gives examples showing that Conjecture 188 is not a weaker statement
than Chvátal’s condition nor Corollary 3.1.2. Figure 4.1 shows specific small graphs
that are guaranteed to have Hamiltonian paths by Conjecture 188, but do not satisfy
Chvátal’s condition, the hypothesis of Theorem 3.2.5 and Corollary 3.1.2. Figures 4.2
31
(a) Example of odd order
(b) Example of even order
Figure 4.1: Distinguishing Graphs for Conjecture 188
and 4.3 diagram how to construct infinitely many such examples.
Figure 4.1(a) displays a connected graph with order nine and degree sequence
(3, 3, 3, 3, 3, 3, 3, 3, 4). This graph fails Chvátal’s condition for k = 4, because dk =
3 ≤ k−1 ≤ n2 −1 and dn+1−k = 3 < 6 = n−k. The annihilation number for this graph
is four and σ = 3 ≤ A(G) − 1. So this graph satisfies the hypothesis for Conjecture
188. Similarly, the graph in Figure 4.1(b) works for Conjecture 188. This graph has
order 12 with degree sequence (4, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 6). So its annihilation number
is six and σ = 5 ≤ A(G) − 1. But if we try to apply Chvátal’s condition we find it
fails for k = 6. dk = 5 ≤ k − 1 ≤
n
2
− 1 and dn+1−k = 5 < 6 = n − k.
At the center of either graph you find a cut vertex. A cut vertex is one that if
deleted will disconnect the graph. These vertices provide the basis for showing that
Theorem 3.2.5 does not hold here. The center vertex in the even order graph also can
be seen as the center vertex of a K1, 3 subgraph. Therefore, this graph is excluded
from Corollary 3.1.2.
The diagram in Figure 4.2 explains how to construct infinitely many examples for
Conjecture 188 of odd order. Here Kr means the complete graph on r vertices. The
32
Kr
Kr
Figure 4.2: Diagram for constructing examples of odd order
Kr+1
Kr
Figure 4.3: Diagram for constructing examples of even order
bold edges work as the join operation between the two graphs they connect. The
regular sized vertices and edges represent regular edges. So the graph represented
here will have 2r + 5 vertices. Figure 4.1(a) is the graph obtained from r = 2. The
degree sequence will consist of one entry equal to 4 and the rest equal to r + 1. Thus
r + 1 = σ ≤ A(G) − 1 = (r + 1) − 1 for all r ≥ 2. Yet for k = r + 2 Chvátal’s condition
fails.
The graphs diagrammed in Figure 4.3 deliver similar results but with even order.
The syntax for the diagram is the same as that described for Figure 4.2. Figure 4.1(b)
is the graph from this diagram when r = 4. For all r ≥ 2 this diagram gives a graph
of order 2r + 4 with a degree sequence consisting of one entry equal to r + 2, one equal
to r, and the rest equal to r + 1. This implies r + 1 = σ ≥ A(G) − 1 = (r + 2) − 1.
However these graphs fail Chvátal’s condition for k = r + 2.
Chapter 5
Maximum Mode and Conjecture
206
5.1
Conjecture 206
The mode of a list is a value that occurs most frequently in the list. Sometimes lists
can be multi-modal, or have more than one mode. This happens when at least two
values occur in the list the same number of times and no other value occurs more
often. This idea can be used to create a few graph invariants.
Definition. The mode of a graph G, mode(G), is the set of integers in the degree
sequence of G that appear in the degree sequence k times and no integer appears
more than k times in the degree sequence.
Definition. The maximum mode of a graph G, m(G), is equal to the maximum
element in the set mode(G).
Sophie produced a conjecture using this invariant which was labeled Conjecture
206 [7]. We will show that it assures that the graph satisfies the hypothesis of Corollary 3.2.10. So Sophie did produce a true conjecture however it does not apply to any
graphs that Corollary 3.2.10 does not.
33
34
Corollary 5.1.1 (Conjecture 206). Let G be a connected graph. If m(G) ≥ 2n(G) −
4 − 2σ(G), then G satisfies Chvátal’s condition.
Proof. Suppose G is a connected graph that satisfies the hypothesis.
Let n =
n(G), σ = σ(G), and m = m(G). Let (d1 , d2 , . . . , dn ) be the degree sequence
for G. Define #(x) to be the number to times the integer x appears in the degree
sequence of G. By hypothesis m ≥ 2n − 4 − 2σ. Since m is the degree of some vertex
in G, n − 1 ≥ m. Thus
n − 1 ≥ m ≥ 2n − 4 − 2σ
(5.1)
n − 1 ≥ 2n − 4 − 2σ
(5.2)
−n + 3 ≥ −2σ
σ ≥
(5.3)
n−3
.
2
(5.4)
Assume that G does not satisfy Chvátal’s condition. Then Corollary 3.2.11 shows
that σ ≤
n−2
.
2
Therefore
n−3
2
≤σ ≤
n−2
.
2
σ can then only be equal to
n−3
2
or
n−2
.
2
Since G does not satisfy Chvátal’s condition, there exists an integer k > 1 such that
dk ≤ k − 1 ≤
Since dk ≥
n−3
2
n
− 1 and dn+1−k ≤ n − k − 1.
2
for all k > 1, then
n−3
n
≤k−1≤ −1
2
2
n−1
n
≤k≤ .
2
2
k is then equal to either
n.
Case 1: n(G) is even.
n−1
k
or n2 . So both σ and k are determined by the parity of
35
Since n is even, σ =
n−2
2
and k = n2 . Since this the k for which Chvatal’s condition
is not satisfied, then
n−2
n
≤ d n2 ≤ − 1
2
2
(5.5)
dn+1−k ≤ n − k − 1.
(5.6)
and
n−2
. Since σ = n−2
and degree sequences are nondecreasing,
2
2
and #( n−2
) ≥ n2 . Substituting σ = n−1
into inequality (5.1), we get
2
2
(5.6) implies d n+2 ≤
2
then d n+2 =
2
n−2
2
the following inequalities:
m ≤ 2n − 4 − 2σ,
m ≤ 2n − 4 − 2
n−2
,
2
m ≤ n − 2.
Since m >
n−2
,
2
then #(m) ≥ #( n−2
)=
2
n
2
and if di = m, then i >
n+2
.
2
Thus
n−2
n
≥m≥ .
2
2
This is a contradiction. So G must satisfy Chvátal’s condition if n(G) is even.
Case 2: n(G) is odd.
Since n is odd, σ =
n−3
2
and k =
n−1
.
2
Since this is the k for which Chvátal’s
condition is not satisfied, then
n−3
n−1
≤ d n−1 ≤
−1
2
2
2
(5.7)
dn+1−k ≤ n − k − 1.
(5.8)
and
36
(5.7) implies d n−1 =
2
#( n−3
)≥
2
n−3
.
2
n−3
.
2
Since σ =
Thus #(m) ≥
By substituting σ =
n−3
2
n−3
2
and degree sequences are nondecreasing,
n−3
.
2
into inequality (5.1) we get the following inequalities:
n − 1 ≥ m ≥ 2n − 4 − 2σ,
m ≥ 2n − 4 − 2
n−3
,
2
m ≥ n − 1,
n−1=m
So every vertex of degree m must be adjacent to every other vertex of G. Since there
are at least
n−3
2
implies, d1 =
vertices of degree m, then
n−3
2
≤ d1 . The fact that d1 ≤ d2 = σ then
n−3
.
2
n−1
n−3
)≥
. Therefore #(m) ≥ n−1
. Which implies n−1
≤
2
2
2
2
d1 = n−3
. This is a contradiction. Thus G must satisfy Chvátal’s condition if n(G) is
2
Now we know #(
odd.
So in all cases Chvátal’s condition is satisfied by the graph G.
5.2
Consequences
This proof made the author realize that when Graffiti.pc was executed for Hamiltonian
path, it must not have “known” [8] Chvátal’s condition. If it had then Conjecture 206
would not have been reported. This is because there would be no example graphs that
would separate Conjecture 206 from Chvátal’s condition. Unfortunately Graffiti.pc
did not rediscover Chvátal’s condition as it did with other conditions. If it had
Graffiti.pc would have removed Conjecture 206 itself.
The strength of Chvátal’s condition makes it a necessary addition to Graffiti.pc’s
37
database. Hamiltonian path was only a test bed for the new Sophie heuristic. But
Graffiti.pc’s results have better defined the line between sufficient and insufficient
for this important NP-complete problem. These conjectures have produced theorems
in this paper and in [10]. So the author believed that a checking with this tool at
Graffiti.pc’s disposal was warranted.
A meeting was graciously scheduled for the discussion of this paper and Graffiti.pc
with Ermelinda DeLaViña. The possibility of a re-testing along these lines was discussed and DeLaViña agreed to do it. So we await the results that are sure to inspire
more interesting exploration into graph theory.
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