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THE COVER nIDEX OF A GRAPH
by
Ram Prakash Gupta
University of North Carolina
Institute of Statistics Mimeo Series No. 562
December 1967
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This research was supported by the Air Force
Office of Scientific Research Contract Bo.
AFOSR-68-1415 and also partially supported
by the National Science Foundation Grant No.
GP-5790.
DEPARTMENT OF STATISTICS
University of North Carolina
Chapel Hill, N. C.
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1.
Introduction.
Let G be a graph (undirected, without loops) with
set of vertices V(G) and set of edges E(G).
Any subset C of E(G) is
said to be a cover of G if each vertex YcV(G) is an endvertex of at
least one edge in C.
The cover index of a graph G, denoted by k(G),
is defined to be the maximum number k such that there exists a decomposition of E(G) into k mutually disjoint sets each of which is a
cover of G.
The lower degree d(G) of G is the greatest lower bound
of the degrees of its vertices.
(For the definitions of the cover
index k(D) and the lower degree d(D) of a directed graph D, see Section
2.)
A graph G is said to be of multiplicity s or simply an s-graph is
no two of its vertices are joined by more than sedges.
In the present paper, we deal with the problem of determining
the cover index of a graph in terms of its lower degree and multiplicity.
For any graph G, we must have k(G) ~ d(G).
and only if d(G) ~ 1.
Also, k(G) > 1 if
O. Ore [9] proved that "for a graph G, k(G) > 2
if and only if d(G) ~ 2 and no connected component of G is a cycle
with odd number of edges".
He then proposed [10] to establish an
analogue of this result for directed graphs.
ing general theorem is proved.
bipartite graph thenk(G)
=
In Section 3, the follow-
Theorem 3.1 "If G is a locally finite
d(G)".
As a corollary to Theorem 3.1, we
then obtain Theorem 3.2 'Tor any locally finite directed graph D,
k(D)
= d(D)".
Evidently, Theorem 3.2 solves a more general problem
than that suggested by O. Ore.
Further, we observe that Theorem 3.2,
and hence, a priori Theorem 3.1, implies a well-known theorem of J.
Petersen [11] and D. Konig [8].
In Section 4, considering the problem for arbitrary graphs (not
2
necess~rily
bipartite) the following theorem is proved.
"If G is any s-graph which is locally finite, then d(G)
Theoram 4.1
2:
keG)
2:
d(G)-s.
The bounds are best possible in the following cases: s > 1 and
-d(G)
where r > 0, m > [r-l]
+ s".
2
In the last section, some remarks on the dU8,lity between the
= 2ms-r
concept of the cover index of a graph, introduced here, and the wellknown concept of the chromatic index of a graph are added.
2.
Definitions and Notations; colorings; The
Conc~pt
of Alternating
Chains.
The method used in proving our results in this paper is based
on the concept of alternating chains introduced bJ' Petersen (1891) in
his invertigations on the existence of subgraphs '\Irhich has been an
effective tool for many problems in graph theory.
In this section,
we first explain the basic concepts and notations which are used
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-,
Two
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vertices are adjacent if they are joined by an edge; two edges are
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throughout this paper, and then define the concept of colorings and
alternating chains.
Definitions and Notations:
An undirected
~'aph
or simply a graph
G is defined by a nonempty set V(G) of vertices aIld a set E(G) of edges
togetherwith a relationship which identifies each edge eeE(G) with an
unordered pair (v,u) of distinct vertices v,u€V(G), called its ~
vertices, which the edge is said to join.
In the following, the letter
G denotes, without any further specifications, a
graph.
(It should
be noted that by definition, we exclude from our c:onsideration graphs
which have 'loops I, 1. e., edges with coincident elldvertices.)
adjacent if they have a common endvertex.
A vertex v and an edge e are
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incident with
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other if v is an endvertex of e.
..---.
As a notational convention, if e is an edge joining the vertices
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e~'l.ch
and u then we write e = (v,u).
e'
= (v',u')
We note that if e = (v,u) and
are two distinct edges, then, unless specifically stated,
we may have v
=
v' or u
= u'
or both v = v' and u = u'.
If no two vertices of a graph G 8xe joined by more than sedges,
s
~
0, then G is said to be of multiplicity s or simply an s-graph.
A graph G is finite if both V(G) and E(G) axe finite; otherwise
G is infinite.
Any subset E' of E(G) defines a partial graph G' of G such that
V(G') = V(G) and E(G') = E'.
Any (nonempty) subset V' of V(G) defines
= V',
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a subgraph G' of G such that V(G t
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partial graph r:£ asubgraph of G is c?"lled a partial subgrp"ph of G.
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E(G')
E(G) and an edge of G
is an edge of Gt if and only if both of endvertices are in Vt
•
A
(It
is implied that every edge of a partial subgraph Gt of G has the same
endvertices in G' as in G.)
Two graphs are said to be edge-disjoint if they have no edge in
common.
A nonempty family {Go' Gl, ... ,Gk _ } of mutually edge-disjoint
l
partial.graphs of a graph G is said to form a decomposition of G if
k.,.l
E(G) =.~ E(G.); we then write
l-v
l
(2.1)
G
= GO
+ Gl + ••• + ~-l
•
A chain in a graph G is a finite sequence (we shall have no
occasion to consider infinite chains) of the form
(2.2)
where (1)
vi
E
V(G), (2)
e.l
€
E(G), and (3) e.l joins v.l - 1 and vl.
for each i=1,2, •.• ,r. (If all the vertices and edges of a graph G can
4
be arranged in a chain then G is itself called a chain.)
~(vo,vr)
is said to connect its 'terminal vertices:'
The chain
and v r ' A
o
graph is connected if every two of its vertices al'e connected by a
chain.
V
The relation of being connected in G is o'bviously an equivalence
relation.
It, therefore, 'partitions' G into a f13LlJlily (G ) of graphs
a
each of which is a maximal connected subgraph of 0 and is called a
connected component of G.
A chain (2.2) is called a cycle if (1) it
has at least one edge, (2) its edges e ,e " .. ,e are all distinct, and
l 2
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(3) its last vertex v r coincides with its first VE!rtex YO'
In a graph G, the degree d(G,v) of a vertex v is the number of
edges that are incident with v,
The lower degree of G, denoted by
d(G), is the greatest lower bound of the degrees of its vertices.
Thus,
the lower degree of a graph is the largest number k such that there
are at least k edges incident with each of its vertices.
A gra.ph is said to be locally finite if the degree d{G.,v) of
each vertex v
€
V(G) is finite,
Trivially, if G is
locally finite,
then its lower degree d{Q) is finite.
Any set of edges C of a graph G, C ~ E(G), is called a cover or
an edge-cover (for the vertices) of G if every vertex v
incident with at least one edge in C.
€
V{G) is
Obviously, if G ha.s an 'isolated
vertex' (a vertex is isolated if its degree is zero) or if d{G)
= 0,
then there can exist no cover of G.
The cover index of a graph G, denoted by keG), is defined as
.
-
follows: keG) = 0 if d{G) = 0; otherwise, keG) is the maximum number
k such that there exists a partition of E(G) into k sets.
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where each of the sets E.(i=O,l, ••• ,k-l) is a cover of G.
1
A partial graph G' of G is called a covering graph of G if
d(G')
21,
or, in other words, if its defining set of edges E(G') is
a cover of G.
The cover index k(G) of a graph G may equivalently be
defined to be the maximum number k such that there exists a decomposition
(2.1) of G where each of the graphs G.(i=O,l, ... ,k-l) is a covering
1
graph of G.
Evidently, if d(G)
= OJ
then no such decomposition of G
exists and in that case we have, by definition, k(G)
we may disregard the trivial case d(G)
necessary, that d(G)
2
=0
= O.
Henceforth,
and assume tacitly, if
1.
A directed graph or briefly digraEh D is defined by a nonempty
set V(D) of vertices and a set A(D) of ~ together with a relationship Which identifies each arc with an ordered pair of (not necessarily
For each arc (y, u),
distinct) vertices which the arc is said to join.
v is called its initial vertex ani u its terminal vertex; both v and
u being called its endvertices.
A digraph D is finite if both V(D) and A(D) are finite; otherwise,
D is infinite.
In a digraph D, the outward degree d+(D,v} of a vertex v is the
number of arcs in A(D) with initial vertex v and the inward degree
d-(D,v) of v is the number of arcs in A(D) with terminal vertex v.
digraph D is locally finite if for each vertex v
~
A
V(D), the sum
d+(D,v) + d-(D,v) is finite.
The lower degree of D, denoted by d(D), is the largest number k
such that d+(D,v)
2 k,
d-(D,v) ~ k for every vertex v
€
V(D).
Obviously,
if D is locally finite, then its lower degree d(D) is finite.
For a digraph D, the notions of partial graph and subgraph are
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defined analogously as for undirected graphs.
~1US,
a partial graph
of D is a digraph D' such that V(D') ::: V(D) and A(D') c:=. A(D).
digraphs are arc-disjoint if they have no arc in common.
Two
A nonempty
family (Do,Dl, •.. ,D _ ) of mutually arc-disjoint partial graphs of a
k l
digraph D is said to form a decomposition of D i:f' A(D) :::\jlA(D ),
i
l=O
and we write D ::: DO + D + ••• + D _ •
l
k l
A set of arcs C of a digraph D, C c:=. A(D), :ls called a cover of
D if for each vertex v «V(D), there is in C at least one arc with
initial vertex v and at least one arc with terminal vertex v.
A
partial graph D' of D is defined as suggested by O. Ore [10], to be a
covering graph of D if d(D') ~ 1 or, equivalent~r, if its defining set
of arcs A(D') is a cover of D.
The cover index of a digraph D, denoted by k(D), is defined
as follows: k(D) ::: 0 if d(D) ::: 0; otherwise, k(D) is the maximum
number k such that there exists a
is the maximum number k such that there exists a partition of A(D)
into k sets each of which is a cover of D.
In the following, we may
p~
objects in this
~aJlter
necessitate the con-
sideration of partitions of the set of edges of ll. given graph.
sider, therefore, for an undirected graph G, an
~l.rbitrary
Con-
decomposition
of its set of edges E(G):
(2.4)
A.
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EO' El , . - -, ~-l; E(G)::: UE i , E:LnE j ::: t.
It is convenient for us to think of each SE!t E. as representing
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color which we denote by the integer i.
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disregard the case d(D) ::: 0 and assume that d(D) ~ 1.
Our
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decomposition of D into k partial
graphs each of which is a covering graph of D; or equivalently, k(D)
Colorings:
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FurthE!r, we say that each
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edge e
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is cFl.lled a coloring or more specifically, a k-coloring (of the edges)
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€
E. is colored with the color i thus obtaining
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the edges of G by means of k distinct colors.
f which assigns to each edge e
of G.
€
~
coloring of
Formally, any function
E(G) an integer f(e)(O,l, ••. ,k-l)
The integers O,l, •.. ,k-l are themselves referred to as colors
and if f(e)
=i
we say that the edge e is colored with the color i or
briefly that e is an i-edge.
It is readily seen that there is a
natural one-to-one correspondence between k-colorings of G and decompositions of E(G) into k sets.
Any k-coloring of G is called admissible if every vertex v
is an endvertex of at least one i-edge for each i
E
= O,l, .•• ,k-l.
V(G)
It
is immediately seen that in a decomposition (2.4) of E(G), each E.l
(i=O,l, •.. ,k-l) is a cover of G if and only if its corresponding
k-coloring of G is admissible.
Thus, the cover index of a graph G is
the maximum number k for which an admissible k-coloring of G exists.
Let f be any k-coloring of G.
c(v),
denote by cf(v) or simply
For any vertex v
€
V(G), we
if no confussion is possible, the set
of colors which are assigned by f to the edges incident with v.
Clearly, cf(v)S (O,l, .•• ,k-l). The difference
(2.5)
Ar(v)
=
(k - Icf(v)
D
is called the deficiency of f at the vertex v.
of elements in the set S.)
Ar(V')=
(2.6)
= t).
denotes the number
For any subset V' of V(G), the sum
\'
v'rt'
is called the deficiency of f
if V'
(lsi
(k-lc f (v)1)
at V'.
(By definition, ~f(V')
=°
The total deficiency of f, denoted by Af(G), is obtained
when the sum (2.6) is eXtended over all vertices of G, 1. e., when
V'
= V(G).
It is clear that a k-coloring f of a graph G is admissible
8
if and only if its total deficiency Af(G)
that k(G)
~
= 0.
Thus, in order to prove
k, where k is some positive integer it is sufficient to
shOv1 that G possesses a k-coloring f whose total deficiency is zero.
The Concept of AIternating Chains:
Any cha.in considered in
the following is assumed to have at least one edge and all of its
terms are to be distinct except that its last vertex may coincide with
one of the vertices preceding it.
Consider an arbitrary k-coloring (k ~ 2) of a graph G And let i
and j be any two fixed colors.
~(vO,vr')
(2.7)
=
A
(vO,el,vl,e2, ••• ,vr_l,er,vr)' r ~ 1,
are alternately colored with i and j, the first one being an i-edge.
An (i, j)-alternating chain (2.7) is called :suitable if it satis-
fies anyone of the following conditions:
v =v .
°
(ii) e r
r'
is an i-edge (respectively, j-edge) and there is no j-edge
(respectively, i-edge) incident vdth v ;
r
(iii)
er is an i-edge (respectively, j-edge) and there is another
i-edge (respectively, j-edge) incident with v •
r
A suitable (i,j)-alternating chain (2.7) is called minimal
if I1(VO'Vt )
= (vO,el'vl , ... ,et,vt )
is not suitable for any t,
1 < t < r-l.
It is easy to observe that if G is finite, then, given
any i-edge e l
= (vO,vl)e
E(G), a (minimal) suitable (i,j)-alternating
chain (2.7) can always be determined.
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chain
in G is called an (i,j)-alternating chain if its edges e ,e , ••• ,e r
l 2
(i)
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(In fact, this can be done by
follovTing along an (i, j ) -alternating chain starting with e l till one
of the conditions (i), (ii) or (iii) is satisfied.)
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Suppose G is a finite graph and we want to show that
G possesses a k-coloring (k ~ 2) which is admissible.
an arbitrary k-coloring f of G.
wise, choosing a vertex V
o
We start with
If Ar(G) = 0, we a.re through.
Other-
> 0 and colors i and j
E V(G) with Af(V )
O
properly, a suitable (i,j)-alternating chain starting with the vertex
v0 is found and the colors i and j interchanged on all edges belonging
to this chain.
Such a step may have to be taken more than once till
we obtain a k-coloring g (say) such that Ag(G) < Af(G).
The assertion
is then proved. by a proper use of finite induction.
3.
The Cover Index of a Bipartite Graph
A granh G is said to be bipartite (simple, C.Birge [I)
contains no cycle with odd number of edges.
if it
Alternatively, G is
bipartite if (and only if) its set of vertices V(G) can be decomposed
joins a vertex v I
E
VI with a vertex VII
E
VII.
These special kind of
graphs are of particular significance in graph theory and play a considerable role in various applications.
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in the following:
into two disjoint sets VI and Vft such that each edge e=(v t, VII) e: E(G)
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The method of alternating chains, as used in this paper, consists
For instance, any (0,1)-
matrix or a family of subsets of a given set can be represented by a
bipartite gra,ph and graph theory becomes a handy tool for sever1'l,l
combinational problems relating to such systems.
Also, we observe
that to any directed graph D, there corresponds, within isomorphism,
a unique bipartite graph G (an arbitrary bipartite graph may not
D
correspond to any directed graph) defined as follows: to each vertex
e
v
E
V(D) there correspond two vertices v t ,v- ft
E
V(GD) and to each arc
(v,u) e:A(D) there corresponds an edge (vt,u ll ) eE(G ).
D
Thus, any
10
problem for directed grapm may be formulated asa problem for bipartite
graphs.
Let G be any graph.
1 respectively.
O.Ore
Cle?"rly, if d(G) = 0 or 1, then keG) = 0 or
(1962) proved that "for
~~y graph G, keG) ~
2 if
and only if d(G) ~ 2 and no connected component of G is ~. cycle with
odd number of edges."
He then suggested [10] to establish an analogue
of this theorem for directed graphs.
The solution to this problem
is provided by the following theorem "for any directed graph D,
keD)
-> 2
if and only if den)
-> 2",
which we can l~OW prove.
Indeed,
this theorem is most conveniently derived as a corollary to the above
theorem of O. Ore.
(The det~.ils of the proof wh:ich are simple, are
omitted.)
We consider below the general problem of d1etermining the cover
index of a bipartite graph which as is easily shl:>wn includes the problem
of determining the cover index of a directed
gr~ph.
for Bipartite Graphs."
and let d(G)
3.1 Let G be a b.ipartite graph which is locally finite
= k.
(3.1)
G
Then, there exists a decompos:ition
= GO
+ G + ••• + G _
k l
l
such that each of the graphs G. is a covering
~
~(Gi)
2
1 (i=O,l, .•• ,k-l).
bipartite grap~ then keG)
Proof:
If d(G)
therefore, d(G)
=0
= k > 1.
grl~ph
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We shall prove
now the following theorem which may be called a "Decomposition Theorem
Theorem
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of G,Le.,
In other words, if G is a locally finite
= d(G).
then the theorem is trivially true.
Evidently, we must havl:! keG)
:s k.
Let,
Hence,
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to prove the theorem, we have to show that k(G)
2:
}t
or equivalently
that there exists a k-coloring f of G which is admissible, i.e.,
6r(G)
We shall infact prove the following lennna which is apparent-
= O.
ly stronger than the above statement.
Lemma 3.1
Let G be a locally finite bipartite graph and let
V' be any subset of V(G) such that d(G, v)
2: k, k 2: 1, for all v
Then, there exists a k-coloring f of G such that Af(V')
Proof:
E
V' .
= O.
The lemma will be proved first for the case in which G
is finite; then, assuming the lennna to be true for all finite graphs,
it will be proved, by using a well-known argument due to D. Konig
and S. Valko [7], when G is infinite (but locally finite).
G is finite:
If k
therefore, that k > 2.
= 1,
the lemma holds obviously.
Suppose,
Consider k-colorings of the graph G.
finite, the number of distint k-colorings of G is finite.
Since
G~
Hence, there
must be a k-coloring f of G such that Af(V') is minimal, i.e.,
A_(V')
< 6.g (V') for any other k-coloring g of G.
~.
-
= O.
6r(V')
Suppose, if possible, that 6. (V') > O.
f
vertex
V
o
E
Then, there will be a
V' such that Ar(v 0) > 0 "Which is equivalent to saying that
there is a color j such that j
fi
cf(v ).
O
Since, now, at most k-l
colors are assigned to the edges incident with
d(G,v
We shall show that
V
o and
by assumption
o) 2: k, there must be a color i such that there are (at least)
two i-edges incident with vO'
Now, we determine a suitable (i,j)-
alternating chain ~(vO,vr) starting with the vertex vo.
Let g denote
the k-coloring of G obtained from f by interchanging the colors i and
12
j on all edges belonging to the chain ~(vO,vr)'
Since the chain is
suitable, it is easily seen that Ic g (v)l2: Icf(v)1 for all vertices
except possibly when v = v.
o
with
In fact, if v r =
YO'
V
We now observe that v
o then
cannot coincide
r
since j ~ cf(v O), ~(vO,vr) would
be a cycle in G with odd number of edges which contradicts the
assumption that G is
~,
bipartite graph.
there were two i-edges incident with
cf(v ) U (j) ~ cf(v ).
O
O
V
Hence, It is seen that, since
o with
re~;pect to f, cg(v
=
Hence, by definition (2.6), it follows that
Ag (V') < Ar (V') which is contradictory to the choice of f.
we must have Ar(V') = O.
G is infinite:
o)
This proves the lemma for
Hence,
order.
Let us first assume that Gf is connected.
Since,
Let Vn (n=1,2, ... ) denote the set of all vertices v such
«
(v l'v2' •• • ,v n) or v is adjacent to some vertex
vi' i=1,2, ••• , n.
Denote by G the
n
1rt;.~
sUbgr~
of Gf defined by the set
of vertices V and let V~ =V'n(v ,v , ... ,v ).
n
n
l 2
Now, it is evident
that the graphs G are finite and d(G ,v) > k for each v
n
n
the lemma holds for finite graphs, it follows
f
n
such that
Ar
(V~)
n
is obviously finite.
'proper' •
coloring f
= 0,
E
V'.
n
Since
G has k-colorings
n
but the number of such k-colorings of G
n
Let any such k-coloring f
IlL
of G be called
n
Now, if n < m, G is a subgraph of G and each proper kn
m
III
of G implies a proper k-coloring f
m
restriction of the coloring of G . f
m'
and we write f
G , G3""
2
tt.~t
n
< fm.
m
n
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of G which is the
n
is called an extension of f
Now, the proper k-colorizllgs of the graphs
imply proper k-colorings of G and si.nce G has only a
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Let vl'v ,v3'.'.' be an enumeration of V(G) in some
2
that either v
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finite graphs.
moreover, G is locally finite, the set of vertices V(G) of G is
enumerable.
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13
finite number of proper k-colorings, there must be one among them
which is restriction of infinitely many proper k-colorings.
fix one such coloring f
l
of this fixed coloring.
of G •
l
Let us
We consider now only the extensions
Each of these implies a proper k-coloring
of G , Hence, by the same reasoning as above, G must hl:'l,Ve a proper
2
2
k-coloring f which is restriction of an infinity of proper k-coloring.
2
We proceed in this manner to obtain a sequence of k-colorings
f l ,f , .•• such that f (n=1,2, ..• ) is a proper k-coloring of Gn and
2
n
Now, we define a k-coloring f of G as follows:
f < f 2 < f < ..•
l
3
for each e € E(G), f(e) = f (e) if n is the first index such that
n
e ~ E(Gn ).
Evidently, f is well-defined and we have Af(V ' )
= O.
If G is not connected, let (G ) be the family of connected
a
components of G.
d(G
a
Let V'a = v'nv(Ga ) for each index a.
Then,
,v) -> k for each v c VIa and since Ga is connected, by what we
have just proved, each of the graphs G has a k-coloring f
a
a
such
that Af (V~) = O. We define a k-coloring f of G as follows: for
a
every e E E(G), f(e) = f (e) if e E E(G ). It is clear that Af(V')=O.
a
a
This completes the proof of the lemma.
The theorem now follows immediately if we take V' = V(G) in
the above lemma.
Remark:
d(G) = k.
Hence, the proof of the theorem is also complete.
Let G be a finite graph Which is bipartite and let
The proof of Lemma 3.1 suggests an algorithm for /'In
actual determination of an admissible k-coloring of G or, equivalently,
a decomposition of G into k covering graphs.
trary k-coloring f
l
of G.
If f
l
We start with an arbi-
is not admissible then a suitable
alternating chain is found as indicated in the proof of the lemma.
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14
By interchanging the colors of all edges belonging to this chain,
we obtain a new k-coloring f 2 of G such that th.e total deficiency of
f
2
is less than the total deficiency of f l by at least 1...
The process
is repeated with the new coloring (each time) till an admissible kcoloring of G is obtained.
Since G is finite, an admissible k-
coloring of G is obtained after a finite number of repititions of
the process. (It may also be noted that the above algorithm is quite
efficient and can be programmed convenie ntly for us e on electronic
computers.)
There are several important consequences of Theorem 3.1 which
we shall state below.
We first obtain the fo1.1.owing theorem which may be called a
"Decomposition Theorem for Directed Graphs".
Theorem 3.2
and let d(D) = k.
(3.2)
Let D be a directed graph which is locally finite
Then, there exists a decomposition
D = DO + D + ••• + D _
l
k l
such that each of the digraphs D. is a covering graph of D,i.e.,
J.
d(D.) > 1 (i=O,l, ••. ,k-l).
J.
-
In other words, if D is a locally finite
digraph then k(D) = d(D).
Proof:
Consider the bipartite graph G corresponding to the
D
given graph D as defined above.
d(G ) = d(D).
D
Clearly, GD is locally finite and
AlSO, it is seen that a set of arcs in D is a cover
of D if and only if its corresponding set of edges in G is a cover
D
of G so that we have evidently, k(G ) = k(D).
D
D
Now, by Theorem 3.1,
we have k(G ) = d(G ) whence the theorem follows immediately.
D
D
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15
...
.l tl+
We say that a digraph D is regular of degree k if
-
d' (D,v) = d (D, v) = k for each vertex v c V(D).
Any partial graph
-.-
-
D' of D is c?lled an r-factor of D if D' is regular of degree r.
If,
in Theorem 3.2, we take D to be regular of degree k, then evidently,
each of the digraphs Di(i=O,l, ••• ,k-l) in the decomposition (3.2)
will be a l-factor of D.
Hence, as a particular case of Theorem 3.2
we obt::dn the following:
Corollary 3.1 Let D be a directed graph which is regular of
degree k.
Then, there exists a decomposition (3.2) of D such that
each of the digraphs D. (i=O,l, ••• ,k-l) is a l-factor of D.
~
Consider an undirected graph G.
The graph G is said to be
regular (homogeneous)C.Berge [lJ) of degree k if d(G,v)
vertex v
E
V(G).
=k
for each
A partial gr?ph G' of G is called an r-factor of
G if G' is regular of degree r.
It is known that if G is a regular
graph of (even) degree 2 k, then the edges of G can be so 'directed'
('oriented'), each edge being assigned a unique direction, that the
resulting digraph Gd , say, is regular of degree k.
Clearl.v, jf a
partial graph Gci of Gd is an r-factor of Ga, then its corresponding
partial graph G' of G is a 2r-factor of G.
Hence, from Corollary
3.1, we obtain further the following 'Tactorization Theorem," due
to Petersen [11].
Corollary 3.2
of degree 2k.
(Theorem of Petersen):
Let G be a regular graph
Then, there exists a decomposition (3.1) of G such
that each of the graphs G (i=O,l, ••. ,k-l) is a 2-factor of G.
i
It may also be noted that the theorem of Petersen implies in
its turn Corollary 3.1.
16
The following result for regular bigartite graphs due to Konig
[8], which is
I'!
special case of a more general theorem of Konig and
P. Hall [6J, cl'ln also be shown to be equivalent to Corollary 3.1 or
the theorem of Petersen and hence, is a consequence of Theorem 3.2.
Corollary 3.3 (Theorem of Konig):
which is regular of degree k.
Let G be a bipartite graph
Then, there exists a decomposition
(3.1) of G such that each of the graphs G.(i=O,l,
••. ,k-l) is 2l-factor
~
of G.
We observe that Theorem 3.1 is apparently stronger than Theorem
3.2 which implies the theorems of Petersen and Konig.
if the converse inplication also holds.
It is not known
Specifically, we may ask the
following question: "Is it true that the theorem of Petersen (or Konig)
implies Theorem 3.11"
(The question was first put to the author by
Professor C. Berge in personal discussion.)
We state below another problem which seems to be of considerable
A digraph D is said to be pseudosymmetr!.£ if i(D,v)=d-(D,v)
interest.
for each vertex v • V(D).
We believe that the following statement
is true.
Conjecture:
Let D be a locp,lly finite directed graph which is
pseu.dosynunetric and let d(D) = k.
Then, there exi:sts a decomposition
of D into k covering graphs each of which is also JPseu.dosynunetric.
4.
The Cover Index of an s -graph
The cover index k(G) of a graph G is the maximum number k such
that there exist k mutually disjoint covers of G.
vertex v
€
If the degree of a
V(G) is k, then clearly, since any cover of G must contain
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17
at leFt,St one edge incident with v, the graph G cannot have more than k
Hence, it is obvious that keG) ~ d(G) where
mutually disjoint covers.
d(G) is the greatest lower bound of the degrees of the vertices of G,
In the previous section, it was proved that if G is a locally finite
bipartite graph then the equality keG) = d(G) holds.
have in general keG) < d(G), as is shown by the following examples.
For any integers s
~
1 and k = 2ms-r where r
~
0, m > [r-l]
2 + s,
we give examples of s-graphs G such that d(G) = k and keG) ~ d(G)-s.
To this end, consider a graph K with 2m+l vertices v ,v , ••• ,v2m in
O l
which each pair of distinct vertices are joined by exactly sedges.
Clearly, K is an s-graph which is regular of degree 2ms.
If k=2ms-r
is even, then, by the theorem of Petersen (Corollary 3.2), K has a
partial graph G which is regular of degree k.
Ie
graph and d(G) = k.
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after simplification, m ~
I.
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However, we may
Obviously, G is an s-
If keG) > k-s+l, then, since any cover of G must
contain at least m+l edges, we have IE(G) I = ik(2m+l) ~ (m+l)(k-s+l) or,
that m > [r;l] + s.
[r~l] + s which contradicts the assumption
Hence, keG) ~ d(G)-s.
AlSO, if k = 2ms-r is odd,
then, it can be easily shown that K has a partial graph G such that
d(G, v ) = k+l and d(G,vi ) = k for i=1,2, ••• ,2m.
O
s-graph and d(G) = k.
Obviously, G is an
As above, it is seen that keG) ~ d(G) - s.
The main result proved below (Theorem 4.1) is that for any locally
finite s-graph G, keG) ~ d(G) - s.
Evidently, in view of the above
examples, this bound cannot be improved in general.
Before proving Theorem 4.1, we first explain some notations and
terminological conventions.
ing of G.
Consider a graph G, and let f be any color-
As earlier, for any vertex v c V(G), cf(v) denotes the set
of colors i such that there is at least one i-edge incident with v.
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18
Further, c~(v) or simply cr(v), r
2:
2, denotes the set of colors i such
that there are at least r i-edges incident with v, and cf(v) or simply
c(v) denotes the set of colors which do not belong to c(v).
In the following, the letters g,f,fO,f , etc. denote colorings
l
Let f be ar..y coloring of G and let IJ.( v, u) be an (i, j) -alternating
of G.
Let g be defined as follows: for an~r edge e
chain in G.
gee)
=j
if e belongs to lJ.(v,u) and f(e)
= i,
=i
if e belongs to J1(v,u) and fee)
= j,
= fee)
€
E(G),
if e does not belong to Jl(v,u).
Then, we say that g is obtained from f by interchanging the colors
on lJ.(v,u).
For the sake of convenience, all suitable
considered below are assumed to be minimal.
a~ternating
chains
Any minimal, suitable
(i,j)-alternating chain is referred to, for brevity, simply as an
(i,j)-chain.
can always be determined.)
We are now prepared to prove the following:
Theorem 4.1:
2:
2:
keG)
cases: s
2:
If G is an s-graph which is locally finite, then
d(G) -s. The bounds are best possible in the following
1, and d(G)
Proof:
= 2ms
- r where r
> 0, m > [r~l] + s.
Let G be a.ny locally finite a-graph, and let d(G)
=k
To complete the proof of the theorem, we have to prove that
keG)
> d(G)
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(As observed earlier, if there is an i-edge incident
with a vertex v, then an (i,j)-chain with v as its initial vertex
d(G)
.-I
- s, or equivalently, that G possesses a (k-s)-coloring
f which is admissible, the rest having been proved above.
prove it only for the case when G is finite.
We
sha~~
The proof can then be
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19
extended to all infinite, locally finite graphs as we did in the case
of Lemma 3.1.
Let, therefore, G be finite.
assertion holds obviously.
If k - s
then the
Hence, we may assume that k - s
start with an arbitrary (k-s)-coloring f of G.
Le., if t\-(G)
.:s 1,
2:
2.
If f is not admissible,
> 0, then we shall show that by suitably IOOdifying the
coloring f, we can obtain another (k-s)-coloring g of G such that
6 g (G) < ~(G). Since G is finite, the assertion will then follow by
finite induction.
The operations used in modifying the coloring f
are as follows:
(I)
Interchange any two colors on all edges of G.
(This is
equiValent merely to renumbering the colors.)
(II)
Interchange colors on an (i,j)-chain t&(v,u) if j
E
and/or there is an i-edge incident with
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any number of times, then Ag (G) <-A_(G).
y
y
which does not belong to
It is clear that if g is obtained from f by using (I) or (II)
M:>reover, we ha,ve evidently,
the following:
Lemma 4.1:
chain in G.
Then, t1 (G)
g
Let f be any coloring of G, and let l£(v,U) be an (i,j)-
Let g be obtained from f by interchanging colors on p(v, u) .
< ~(G) if
j • Cf(v) and there is an i-edge incident with
I
LJ-
,.'
which does not belong to ~(v, u).
We now proceed with the proof of the inequality k(G)
Let f be any (k-s)-coloring of G which is not admissible.
2:
d(G) - s.
Then, there
V(G) such that ~(vo) > 0 or equivalently, c(vo) f t.
Suppose that we also have c3 (v ) f$. Let i E c 3 (v ), and let j E: C(V )
o
is a vertex V
o
E
o
I.
c(v),
L'
~(v,u).
I.
We
Consider an (i,j)-chain lJ.(vO'u) and let g be obtained from f by
interchanging the colors on lJ.(vO'u).
Since
o
::>0
i
€
c 3 (v0)' it is seen that there is an i-edge inddent with v 0 which
does not belong to p(v0' u) so that by Lemma 4.1, 1ife have Ag(G) < A.r(G),
as required. Hence, we may assume that c 3 (v0)
=k
It is observed that since d(G)
2
is k - s, for any vertex v, c (v)
Now, let i
l
2
c (v 0) and e 1
C
f'rom (b), i
n
2
c ("0) = t, and (c)
l
<f
t.
and the number of' colors used
2
1
t, and if' c 3 (v)
=
(v 0' vI) be an:r iI-edge.
2
assume, if possible, that (a) c (vI)
?(b) c (vI)
=
3
C
n -c
(vI)
then Ic (v>l.2: s.
Let us
2
(v O) = t J, Le., c (vI) c.=. c(vO),
=
t hold simultaneously.
2
Ic (v ) I .2: s.
l
2
c (v ) and f'rom (c),
l
= .,
Then,
Since there 8,re at
roost s-l edges, other than the il-edgese , joining V o and VI' it f'ollows
l
2
from (a) that f'or some color i e: c (v ), there IlIUl3t be an i?--edge
2
l
e
2
= (v0' v2)'
say, such that v 2
2
possible, that c (v 2)
c 3 (v 2 )
=
t.
Then, clearly,
2
::>
and c (vI) U c (v?-)
i
2
c c (v )
1
3
that v
Ic(v 0)
n c(v0) =
s. c(v 0).
u c 2 (v 2 )
I :s k-s -1,
vI·
Let us assume f'urther, if'
2
2
t, c (v ) n c (v ) =t f'or t = 0,1, and
t
2
2
2
2
c (vi) U c (v ), Ic (v ) u c2(v2)1.2:2S
2
l
Hence, as above, fOJ:" some color
there must be an i -edge e
~ {v 0' v l' v 2 }.
3
~f.
f
3
3
=:
(v ,v ), say, such
O 3
Continuing the process, it is evident that since
there must exist an index m, 1:S m
we C8n f'ind it-edges e t
= (vO,v t ),
t
= 1,2, ... ,m
:s k-s-l,
such that
satisfying the f'0110;W-
ing conditions:
I)
(ii)
(iii)
i
l
€
c (v )' and for each t, 2
O
:s t :s m,
there is a (unique) iniex
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21
(iv)
(v)
2
c (vm_1 ) are mutually disjoint,
C2 (VO)' C2 (V1 ), •.• ,
c3 (v
3
o) = c (v 1 )
= c 3 (vm_1 )
= •.•
=
+,
and further we have one of the following cases:
(1)
c2-(v ) n -c(v )
m
O
(2)
c (vm)
n
2
c (vm)
n c2 (vt )
c 3 (vm)
f +.
(4 )
2
f +,
2
c (v 0)
or
f +,
f
or
+ for some t, 1 ~ t ~ m-1, or
In each of the above cases, we shall now obtain a (k-s)-co1oring
g of G such that Ag(G) < ~(G).
Case (1).
2
c (ym) n c(vo)
f t.
2
Let j • c (vm) n c(vo)'
changing the colors j and 0, if necessary, we may take j=O.
InterLet the
Ie
new coloring of G be denoted by fO'
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is evident from conditions (ii) and (iv) that there is a (unique)
I.
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I
all the conditions (i) to
(v)
(Obviously, Af (G) = Ar(G), and
o
are satisfied with respect to f
o.) It
sequence of vertices
(4.2)
Now, considering the sequence of vertices (4.2), we proceed as
follows:
with
Vo
Sin::e i
2
• c (yo)' there is an i-edge ep' , say, incident
P1
P1
1
such that e'
P1
f
e
P1
= e .
1
such that the i-edge e
and one of the i-edges e
and e' do
P2
P2
P1
P1
P1
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22
do not belong to it.
(Since i
~(vO,u)
In fact, let
found.
and ~'(vO,u') be an (i
PI
,i
P~
P?
€
2
c (v
be an (i
PI
,i
), such a chain can al,vays be
PI P2
)-chain with first edge e
)-chain with first edge e'.
PI
Then, it is
PI
easily checked that one of these chains must satisfy the requirement.)
Using (II), interchange the colors on ~(vO,uI)' and let f l denote the
coloring of G so obtained. Evidently, Af~G) ~Af(G), and with respect
2
2
to f l , we have i
• c (v )' i
€ c (v = ) for j = a+l,a, ••. ,3, and
O
P2
Pj
Pj - l
o • c
2
(v ) n c(v o)'
Since i
m
say, incident with
(i
P2
,i
P3
c
2
such that e'
o
P~
(v O)'
f
e
there is an i-edge e'
P2
P2 '
• As above, consider an
P2
)-chain ~(vO,u2) such that the i-edge e
and one of the
P
P3
3
i-edges e
P
V
P2
€
2
P
and e'
P
2
do not belong to --(v ,u ).
~c
O 2
2
Using (II), inter-
change the colors on "-2(vo'~) and let f 2 denote the coloring so obEvidently, Af (G) ~ Af (G) ~ A.r(G), and -with respect to f ,
2
2
I
2
2
we have i
£ c (v )' i
• c (v
) for j = a+l,a, ... ,4, and
O
P3
Pj
Pj - l
tained.
o
€
c2 (v m)
n -(
c vO).
Proceeding in this manner, i t is obvious that we
can obtain, using (II) a-2 times more, a coloring f
a
f
~j (G) ~ Af(G), and with respect to fa' we have i.
Pa +l
a
=-
?
OEC(V
Since i
V
o
-
Pa +l
Pa +l
)nc(v
E
o)
(where e
Pa +l
=
(vO,v p
a+l
of G such that
E
2
c (v o)'
) is an i-edge).
Pa +l
2
c (v )' there is B.n i-edge e'
, say, incident with
o
Pa +l
Pa +1
such that e'
Pa +l
f
e
Pa +l
.
Now, consider an (i
Pa +l
,O)-chain p(vo'u),
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23
say, with first edge e'
• Using (II), interchange the colors on
Pa+l
~(vO,u) and let g be the coloring so obtained. Since 0 € c2 (v
)
Pa +l
and ~(vO,u) is minimal, it is seen that e
Pa+l
cannot belong to ~(vo,u)
so that by Lemma 4.1, we have Ag(G) < ~ (G) S~(G), as required.
a
Case (2).
by
(v), c3 (vo )
= t, so that there are exactly two i-edges e and e',
say, incident with vO.
~(vO,u), say.
Let j
€
c(v ) and consider an (i,j)-chain
o
Using (II), interchange the colors i and j on ~(vOu),
and then, interchange i and 0 on all edges of G.
Let 1'0 denote the
(.'
coloring so detained.
If one of the edges e and ;i-I does not belong
to ~(vO,u), then we take g = 1'0 and by Lemma 4.1, we evidently have
Ag(G) < Ar(G).
If both e and e' belong to p(vO,u)(or, equivalently,
if u =v ), then it is observed that with respect to 1'0' we have
O
2
2
o E c (v ) n c(vo )' and i € c (v
), where e
= (vO' v ) are
m
Pj
Pj - l
Pj
Pj
i
Pj
-edges, for j = a+l,a, ..• ,l.
with the sequence of vertices
Hence, as in case (1), proceeding
(4.2),
we can obtain g such that
Ag(G) < ~(G), as required.
Case(~.
i
€
c2 (vm)
2
C
(vm)
n c2 (vt ).
n C2 (vt ) f •
for some t: 1 < t < m-l.
Let
From conditions (ii) and (iv), there is a (unique)
sequence of vertices
vt=v
it = i
~+l
€
qb+l
, v , .•. ,v =Vl,v =VO(t=qb+l > qb >... ~=O) such that
qb
ql
~
2
c (v
qb
), i
qb
€
2
c (v
qb-l
), ••• , i
~
€
2
c (v
2
1)' i l =i € c (v ) .
qql
~
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24
Now, in view of the cases (1) and (2), we may assume that ie c(v )
0
2
and i ~ c (v o) so that there is one and only one i-edge e, say, incident
with v.
f
From (ii) and (iv), it is easily seen that e
equivalently, i
f
i r for any r
= 1,2, .•. ,t.
e r or,
In particular, i
f
i
any r = b+l,b, ... ,1.
Now, let us first assume that we also have i
f
f
i
~
for
(or,
Pr
) for any r = a+l,a, •.. ,1. Let j E c(v ), and
o
Pr
consider an (i,j)-chain p(vO'u), say. (Obviously, u f v O.) Using (II),
equivalently, e
e
interchange the colors on ~(vo,u), and then interchange i and 0 on all
edges of G.
Ar
o
Let f
(G) SAr(G).
O
denote the coloring so obtained.
Now, if u
respect to f O' we have 0
e
Pr
= (vO'v
f
v
m
=v
2(v)
m
E C
P~+l
, then it is seen that with
nc(v)
O'
~'are i
-edges, for r
P';~
Pr
If u
2
that with respect to f O' we have 0 E c (v t )
where e
qj
= (vO'v )
qj
are i
qj
and i
Pr
2 (v
E C
= a+l,a, •.. ,l.
case (1), we can obtain g as required.
Evidently,
= vm'
Pr - l
) where
Hence, as in
then it is seen
2
n c(vo)'
and i .E c ( v ) ,
qJ
qj-l
-edges, for j = b+l,b, ••. ,l.
Hence,
we can obtain g as required.
Let us now assume that i
for some r
i
f
i
Let j
Pl
E
•
= a+l,a, ••• ,1.
Let i
c(v0)'
may take j
= 0,
=i
Pr
(a+l
=i
~
(or, equival,~ntly, that e
=e
Pr
)
2
Since, by assumption, :t~ c (v ), clearly
O
> r > 2), so that i
-
-
Pr
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_I
evidently, as in case (1), proceeding with the sequence of vertices
(4.3),
.'I
2
c (v )
m
n
c 2 (v
).
Pr - l
Interchanging the colors j and 0, if necessary, we
and denote the new coloring of G by f '
O
Now, it is
evident that we can obtain a coloring f _ of G, proceeding as in case
r 2
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(1), such that ~
(G) ~Af (G)
r-2
we have 1.
).
c2V(
O' 1
E
Pr - l
=
(vO'v
) are i
E
c'2(v )
n
Pj
m
Pj
E
Pj
c2 (v
).
Pr- 1
Pr - l
Pr - l
and with respect to f r _2 ,
) for j
c ~( v
Pj - l
-edges for j
v 0 such that e l f e
with v
.
Pr - l
= ~(G),
0
= a+l,a, •.. ,r,
where
a+l,a, •.. ,r-l, and
=
Now, let e I
be an i
-edge incident with
Pr - l
Pr - l
, and let e I, e tI be any two i
~'(V
Consider ( i , i
)-chains
Pr Pr - l
Pr-l
with first edge e' and e tl , respectively.
-edges incident
Pr
,U') and JltI(v
,uti)
Pr - l
Since both of these chains
are minimal, it is seen that at least one of them, say JI, (v
,U
'),
Pr - l
does not contain the i-edge e •
Pr
Pr
If u,
f v
~I(Vp
using (II), first interchange the colors on
to O.
change the color of e
f
and u'
Pr - l
YO' then,
,u'), and then
r-l
Let g be the coloring so obtained.
Pr - l
It is easily verified that Ag (G) < Ar
U
I
= v r- l'
is an (i
e
Pr
(G) ~ Af(G), as required.
= (v o'
then it is seen that J.lr_l(vO,u ' )
Pr - l
,i
Pr
If
r-2
ep
,
tl'{v r_l'u'»
r-l
)-chain which does not contain the edges e
Pr - l
and
Using (II), interchange the colors on ~r_l(vo'u') and let f r _l
•
denote the coloring so obtained.
And, if u I
= V 0'
then ~ I (v
Pr - l
,UI)
contains the edge e '
, but does not contain any of the edges e
,
Pr - l
Pr - l
e
and e tI.
Using (II), interchange the colors on
Pr
let f
r-
1 denote the coloring so obtained.
~ I (v
, U I), and
Pr-l
In either case, it is seen
26
that Af
i
e
Pr
Pj
€ C
r-l
2
(G).:s ~(G), and with respect to f r- l'
n
(v m)
= (vo'v P
2
c (v 0), i
) are i
j
Pj
€
P,i
lev
Pj - l
-edges for j
) for j
= a+l,a, ... ,r+l,
= a+l,a, ... ,r.
we have now a situation similar to case (2).
have
life
where
It is evident that
Hem~e,
we can obtain g
satisfying 6 g (G) < ~(G), as required.
3
Case
(4).
) f t.
_
__
_ _c_(v
_m___
Let i
E
c 3 (v m).
From the above cases
2
(1), (2), and (3), we may assume that ie'c(v ), i t c (v ) and
O
o
2
i . c (v ) for any t = 1,2, ••• ,m-l. Hence, there is one and only one
t
i-edge incident with v 0' and i
j c c(v ).
f
it for any t
Consider an (i,j)-chain ",(vO,u).
o
= 1,,2, .•. ,m.
Let
Using (II), interchange
the colors on p(v 0' u), and then, interchange i anel 0 on all edges of
G.
Let f
o
denote the coloring of G so obtained.
It is clear that
A (G) .:s 6r(G), and furthermore, with respect to :f'0 we have
f
o
oe
2
c (v )
m
n c 2 (v 0),
i
Pj
€
edges for j = a+l,a, •.• ,l.
2
c (v
Pj - l
), where e
Pj
= (vO,v
Pj
) are i
Pj
Hence, as in case (1)" we can obtain g
as required.
Thus, we have shown that starting with any (k-s)-coloring f of g
such that Ar(G) > 0, we can obtain a (k-s)-coloring g of G satisfying
Ag(G)<ty(G). The proof of the theorem is n01v completec1 by finite induction.
A graph is said to be linear if no two of its vertices are joined
by more than one edge.
As an important specil3.1 case of Theorem
4. 1, we have the following:
Theorem
4.2.
If G is a linear graph which is 10caJ.ly finite,
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then k(G)
= d(G)
or k(G)
For any integer k ~ 2, there exist
= d(G)-l.
linear graphs G such that d(G)
=k
and k(G)
= d(G)-l.
It may be remarked that the proof of the inequality k(G) ~ d(G)-s
in Theorem 4.1 is algorithmic so that given any finite s-graph G we
can find a partition of its set of edges into d(G)-s sets each of which
is a cover of G.
Remark:
We state here without proof that the Theorem::3.1 and 4.1
remain valid if we replace the condition of locally finiteness by the
=k < •
more general conditions: (a) d(G)
and (b) each connected
component of G has an enumerable number of vertices.
The above
theorems may hold for arbitrary graphs satisfying only condition (a).
However, no proof or counterexample to it is known to us.
5. Duality between Cover Index and Chromatic Index of a Graph
Let G be an undirected graph.
a matching of G if each vertex v
edge in M.
€
Any subset M of E(G) is called
V(G) is incident with at most one
The chromatic index of G, denoted by q(G), is defined to
be the minimum number q for which there exists a partition of E(G)
into q sets each of which is a matching of G.
Equivalent~,
the
chromatic index q(G) of G may be defined, as usual, to be the minimum
number q for which there exists a q-coloring f of G such that for
each vertex v E V(G), there is at most one i-edge incident with v for
i
= O,l, ••• ,q-l.
It may be observed that the concept of the cover
index of a graph, as introduced in this paper, is apparently dual to
the concept of the chromatic index of a graph, and it is interesting
to compare the knmm results concerning these two concepts.
We may,
28
therefore, add the following remarks:
(1) We proved (Theorem 3.1) that if G is
l'l.
(locally finite)
bipArtite graph, then keG) = d(G), and as is well-known, vie also have
q(G)
= d(G)
where d(G) is the least upper bound of the degrees of the
vertices of G.
= d(G)
In fact, the equality q(G)
is equivl'l,lent to
the theorem of Konig, Corollary 3.3, (see, for instance, C. Berge [1],
= d(G).
p. 99), and hence is implied by its counterpart keG)
It is not
known, however, if the converse implication also holds.
(2)
d(G)
We know that "if G is any locally finite s-graph then
:s q(G) :s d(G)
+ s; the bounds are best possible in the following
_'.. (, .1,
cases: s
2
1 and d(G)
= 2ms-r
where r
]~+l
2
:.:
0, m > ~~J.
Clearly, the
I'l,bove theorem and Theorem 4.1 are counterparts of each other and it
is tempting to ask if anyone of
iTl
the~ CM
be derived from the other.
-
(The inequality
q(G) < d(G) + s
.
Very little is known in this respect.
was initially proved by V. G. Vizing [12], and
the author independently.
(3)
keG)
=
See, e.g.,
l~Lter
,.,
[4,5J.)
-
= d(G).
Unfortunately, as one might
expect, such a relationship does not hold in genElral.
if G is the graph of Figure 1, then keG)
For instance,
= d(G) 'but q(G) = d(G) +1,
and if G is the graph of Figure 2, then q(G) :: d(G) but keG)
In general, for any s
2
1 and any fixed r, 1:S r
examples of s-graphs G such that keG)
= d(G)
= d(G)
= d(G)
+ r
d(G) whereas
- r. Hence, one can only ask for SOmE! criterion under
which both the equalities keG)
ous1y.
:=
= d(G)-l
one can construct
where q(G)
and also, examples of s-graphs G such that q(G)
keG)
:s s,
= d(G)
and q(G)
Also, it would be desirable to have
= d(G)
crite~ria
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rediscovered by
It can be easily seen that if G is a regular graph, then
d(G) if and only if q(G)
I
hold simu1taneunder which the
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equality k(G) = d(G) or q(G) = d(G) holds.
Figure 2.
Figure 1-
k(G) = d(G), q(G) = d(G)
+ 1
q(G)
=
d(G), k(G) = d(G) - 1
30
ACKNOWLEDGEMENTS
}bst of the present research work was carried at the Tata
Institute of Fundamental Research, Bombay and is embodied in the thesis
of the I'l.uthor [5].
It is a grep,t pleasure to express my sincere
gratitude to Professor S. S. Shrikhande for supervising the research
work and providing steady encouragement in the :preparation of the
mI'l.nuscript.
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REFERENCES
[1]
C. Berge, Theory of Graphs and its Applications, Methuen and
Co., London (1962).
[2]
R. P. Gupta,
A Decomposition Theorem for Bipartite Graphs
(Abstract), Theory of Gre,phs, International Symposium,
Rome, July 1966, pp. 135-136.
(3)
R. P. Gupta,
A Theorem on the Cover Index of an s-gr!'l,ph
(Abstract 637-14), Notices of the Amer. Math. Soc., Vol.
13 (1966), p. 714.
[4]
R. P. Gupta,
The Chromatic Index and the Degree of a Graph
(Abstract 66T-429), Notices of the Amer. Math. Soc.,
Vol. 13 (1966), p. 719.
[5]
R. P. Gupta,
Studies in the Theory of Graphs
(thesis), Tata
Institute of Fundamental Research, Bombay (1967),. Chapter 2.
[6]
P. Hall,
On Representations of Subsets, Jour. London Math.
Soc., Vol. 10 (1934), pp. 26-30.
[7]
D. ~nig and S. Valko,
Uber Mehrdentige Abbildungen von Mengen,
Math. Annalen, Vol. 95 (1926), pp. 135-138.
[8]
D. KOnig,
Theorie der endlichen und unendlichen Graphen,
Akad. Verl., Leipzig (1936), pp. 170-175.
[9]
o.
Ore,
Theory of Graphs,
A-M-S. Colloquium Publications 38
(1962), Theorem 13.2.3, pp. 208-210.
[10]
o.
[11]
J. Petersen, Die Theorie der Regularen Graphs, Acta Math.,
Ore,
loco cit., Problem 4, p. 210.
Vol. 15 (1891), pp. 193-220.
32
[12]
V. G. Vizing,
On an estimate of the chromatic class of a p-graph
(Russian), Discrete Analiz, No.3 (196!J.), pp. 25-30.
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