A study of the total chromatic number of equibipartite graphs

DISCRETE

MATHEMATICS

ELSEVIER Discrete Mathematics 184 (1998) 49-60

A study of the total chromatic number

of equibipartite graphs

B o r - L i a n g C h e n a'* C h u n - K a n C h e n g b, H u n g - L i n F u b, K u o - C h i n g H u a n g c

a Department of Mathematics, Tunghai University, Taichuny, Taiwan, R.O.C.

b Department of Applied Mathematics, National Chiao Tung University, Hsienchu, Taiwan, R.O.C. c Department of Applied Mathematics, Providence University, Taichung Hsien, Taiwan, R.O.C.

Received 17 June 1996; received in revised form 2 April 1997; accepted 14 April 1997

Abstract

The total chromatic number z t ( G ) of a graph G is the least number of colors needed to color the vertices and edges of G so that no adjacent vertices or edges receive the same color, no incident edges receive the same color as either of the vertices it is incident with. In this paper, we obtain some results of the total chromatic number of the equibipartite graphs of order 2n with maximum degree n - 1. As a part of our results, we disprove the biconformability conjecture. @ 1998 Published by Elsevier Science B.V. All rights reserved

1. Introduction

A total coloring o f a graph G is a mapping ~: V(G)UE(G)--+C such that no incident or adjacent pair o f elements o f V ( G ) U E ( G ) receive the same color. Thus a total coloring o f G incorporates both a vertex coloring and an edge coloring o f G, and satisfies the additional condition that no vertex receives the same color as an edge incident with the vertex. The total chromatic number J(t(G) is the least value o f ICI for which G has a total coloring.

A well-known conjecture o f Behzad [1], and independently o f Vizing [8] is that A(G) + l <~zt(G)<~A(G) 4. 2. The lower bound here is easy to see, but whether the upper bound holds is still unknown. This is also called the total coloring conjecture (TCC).

If the conjecture is proved to be true for a class o f graphs, then the graphs G having z t ( G ) = A ( G ) + 1 are type 1 graphs, and the other graphs are type 2, i.e., zt(G) = A(G) 4- 2.

* Corresponding author. E-mail: blchen@s867.thu.edu.tw.

0012-365X/98/$19.00 Copyright @ 1998 Published by Elsevier Science B.V. All rights reserved PH SOO12-365X(97)OO160-X

50 B.-L. Chen et al./ Discrete Mathematics 184 (1998) 4 9 4 0

In [7], Rosenfeld proved that a bipartite graph satisfies TCC, which is also immediate from the result of K6nig which states that a bipartite graph is of class 1 [3]. Thus we can study the classification problem o f bipartite graphs. The following results are known.

Theorem 1.1 (Behzad et al. [2]). A complete bipartite graph Km, n is type 2 if and only if m=n.

Theorem 1.2 (Hilton [6]). Let J be a subgragh of K,,,, e(J)--IE(J)I, and re(J) be the maximum size of a matching in J. Then xt(K,.n\E(J)) = n + 2 if and only if e(J) + m(J)<.n - 1.

In what follows we shall focus on the bipartite graph G = ( A , B ) where IAI =

]BI

= n . Such a graph is also called an equibipartite graph. It can be seen that Theorem 1.2 is mainly concerned with equibipartite graphs of order 2n with maximum degree n. In this paper, we shall study the equibipartite graphs of order 2n with maximum degree n - - l .

For equibipartite graphs, it is convenient to present a total coloring by using an array with its sideline and headline. Let G = (A,B) be an equibipartite graph of order 2n where A = {x],x2,...,x,} and B = {yz,Y2,... ,y,}. If rc is a total coloring of G, then it has an n × n array M such that M(i,j) = g(xiYj) where xiYj E E(G), and the sideline (and headline) of M represents the vertex coloring of A (and B) with respect to re. Let M* be M with its sideline and headline. Then M* will be referred to as a total coloring array of G. (Fig. 1. is an example.)

Note that if G is type 1, then M will be a partial latin square of order n, further- more each row including the sideline contains distinct elements, so does each column including the headline. Clearly, in order to be a total coloring array of an equibipartite graph, M* has to satisfy some further conditions on vertex coloring.

x 2 ~ Y 2

x 3 ~ Y 3

G. x4 ~

y~

x s ~ ¥ 5

X6 ~

76

X T ~ - - - " " ' ~ - - " " - ~ Y 7 2 3 4 5 6 7

7 l tnlrl714ts

4 [71SI5i~i2J7

16

4 L z l z f r i n l 3 r s i 7

5 }~1617! I {~II2 t4

5

I I

14!31[]t712

614151217111N[3

7 51417[2t613!~

Fig. 1,

B.-L. Chen et al./Discrete Mathematics 184 (1998) 49~50 51 A graph G is biconformable if G is equibipartite and G has a vertex coloring

~o: V(G)---~ {cl,c2 . . . ca(6)+l} such that the following conditions hold: (i) d e f ( G ) =

~ c v ( G ) ( A ( G ) - degc(v))>~ ~i(=~ )+l ]ai - bi], and (ii) IV<~(A\Aj)I >~bj - aj and

I V < ~ ( B \ B j ) [ > ~ a j - bj for each j E { 1 , 2 . . . A(G) + 1}, where

Iv<~(s)l

is the number of vertices in S C_ V ( G ) which have degree less than A(G), Aj =-qg-l(cj)hA, B j = < p - t ( c j ) N B , a j = l A j l , and b j = l B / I .

The following result shows that biconformability is a necessary condition for the equibipartite graph to be type 1.

L e m m a 1.3. Let G = ( A , B ) be an equibipartite graph. I f G is type 1, then G is biconformable. Equivalently, if G is not biconformable, then G is type 2.

Proof. Since G is type 1, there exists a total coloring n which uses A ( G ) + 1 colors. Let these colors be cl,c2,... ,CA(G)+I. Therefore, n Iv(G) is a vertex coloring of G which uses the colors cl,c2 .... ,c~cc)+l. For each color cj, let aj and bj be the number of vertices in Aj and Bj, respectively, which are colored with cj. Since G is a bipartite graph, for each j = 1,2 . . . A(G) + 1, there are at least ]aj - b j ] vertices of G in which the color cj is missing on the edges which are incident with these laj - b j ] vertices. This implies that there are at least laj - b j l vertices which have deficiency one. Thus def(G)>~ ~/~16)+1 [ai -- bi], IV<~(A\Aj)I >~bj - aj and IV<a(B\Bj) I >~aj - bj follow

by the same reason. This concludes the proof. []

For the equibipartite graphs of order 2n with maximum degree n, we can obtain a necessary and sufficient condition for being biconformable.

L e m m a 1.4. Let J be a subgraph o f Kn,n which has at least one isolated vertex. Then G = Kn.n\E(J) is biconformable if and only if e(J) + m ( J ) >~ n.

Proof. If G is biconformable, then there exists a vertex coloring q~: V(G)---~ C =

{cl,c2 . . . C~(a)+l} such that def(G)~> ~-'~__(~ )+l l a i - bi[. Let t be the number of independent edges of J in which two end vertices have the same color. Clearly,

X-,~(G)+l

t <~ m(J). Also, z_~i=l l a i - bil >~ 2 n - 2t. By the fact that J contains an isolated ver- tex, we have A ( G ) -- n and 2e( J ) -- def(G) >~ ~A<_~)+I ] ai - bi I >~ 2n - 2t >~ 2n - 2m( J ).

Hence, e(J) + m(J)>~n.

Conversely, by Theorem 1.2, G is type 1 and then by Lemma 1.3, G is bicon- formable. [~

We note here that if G =Kn,n\E(J), where J contains at least one isolated vertex, then by Lemma 1.4, G is type 1 if and only if G is biconformable. But this conclusion is not true in general. A well-known example, the M6bius band of order 14, MI4

(Fig. 2) is biconformable and it is a type 2 graph. In [5], Hamilton et al. posed the so- called biconformability conjecture in order to obtain a clear picture for the classification of bipartite graphs with respect to total coloring.

52 B.-L. Chen et al./Discrete Mathematics 184 (1998) 49 60

Fig. 2.

Conjecture 1.5 (Biconformability Conjecture). Let G be a bipartite graph with A(G) >~(3/14)(IV(G)I + 1). Then G is type 2 if and only if G contains an induced equibi- partite subgraph H with A ( H ) = A(G) which is not biconformable.

In this paper, we study the total coloring of equibipartite graphs of order 2n with maximum degree n - 1 and we obtain some sufficient conditions for the graphs to be type 1. Clearly, one of the requirement is 'biconformable'. As long as the bi- conformability itself is not enough to ensure that the graph is type 1, then there is a possibility of obtaining a counterexample to Conjecture 1. We shall mention a class of counterexamples in Section 4. Finally, following our results, we pose a conjecture in the direction of solving the classification problem of the equibipartite graph of order 2n with maximum degree n - 1.

2. The basic lemma

It is easy to see that if H is a subgraph of G such that A ( H ) = A ( G ) and G is type I, then H is also a type 1 graph. In other words, if we delete some edges from a type 1 graph G without changing the maximum degree, then the graph obtained is also type 1. Therefore, it suffices to study the maximal one which has degree A(G).

A vertex is called a major vertex of G if the degree of this vertex is A(G), and the vertex which is not a major vertex is called a minor vertex. A graph is maximal

if all the minor vertices are mutually adjacent. Now the following lemma is easy to see.

L e m m a 2.1. I f G is a maximal subgraph of Kn,n with A ( G ) = n - 1. Then J=Kn, n \E(G) is a (vertex) disjoint union of stars.

Proof. Suppose not. Since A ( G ) = n - 1 , the degree of each vertex of J is at least one. Therefore J is a spanning subgraph of Kn, n. I f J contains a cycle, J must be an even cycle, and hence there exists a pair of minor vertices in G which are not adjacent. This is not possible for a maximal graph. Therefore J is a spanning forest. Now if there exists a component of J which is not a star, then there are two adjacent vertices in the component which are of degree at least two. This implies that in G, there are

B.-L Chen et aL /Discrete Mathematics 184 (1998) 49-60 53

xl,

Yz

x 3

73

X4

7 4

X5

¥S

x 6

o

~

7 6 Fig. 3. (3;2,23; 1) graph.

two minor vertices which are not adjacent. Again it is a contradiction. Hence we have the proof. []

Let G = ( A , B ) = K ~ , , \ E ( J ) be a maximal graph with A ( G ) = n - 1. Since J is a disjoint union o f stars, hence denote J by an (s + t + 1)-tuple (ml,m2,...,m~; nl,n2 . . . nt; r)n where mi, i = 1 , 2 , . . . , s , is the degree o f xi CA such that d e g j ( x i ) ~ 2 , nj is the degree o f yj CB, j = 1 , 2 , . . . , t , such that degj(yi)>~2, and r is the number o f independent edges in J . Without loss o f generality, we may assume that ml t> m2/> . . .

s

>~ms, nl>~n2>~...>~nt, and d e g j ( x ~ ) = d e g j ( y k ) = l for each k>~t+ff~i=lmi=

s + ~tj=l nj. Therefore we also have

degj (Xs + 1 ) = degj (Xs+2) . . . degj (Xs+Z, ' n, ) = 1

degj(yt+l ) = degj(yt+2) . . . degJ(Yt+~L, m~ ) = 1,

s

and n = t + r + ~ i = l mi = s + r + ~--Jj=l n). For clarity, we give an example in Fig. 3. The following result characterizes the biconformability o f the maximal equibipartite graph o f order 2n with maximal degree n - 1.

Proposition 2.2.

Let J = ( m l , m 2 , . . . , m s ; n t , n 2 .. . . . nt;r)n and G=Kn, n\E(J). Then G

is biconformable if and only i f

1

j=l + r ~ n . (1)

s + ,-1 ma rn, + t Jr- t

Proof. I f G is biconforrnable, then there exists a vertex coloring ~p using the colors cl, c2 . . . cn which satisfies the biconformability. Let N j ( x ) denote the neighbor o f x in J . I f x E A (resp. B) is a center o f a star o f size at least 2, and ci is a color which occurs in N j ( x ) , then clearly only x and the vertices which in B (resp. A) can be colored with ci. On the other hand, if two centers o f A (resp. B) are colored with ci, then ci cannot occur in B (resp. A). This implies that for each star o f size l ~ 2 ,

n

54 B.-L. Chen et al./Discrete Mathematics 184 (1998) 4940

star contains a vertex of degree one which uses a common color with the center or

n

not. Thus by the fact that def(G) = y'~isl (mi - 1) + ~-~l=l (ni - 1) >~ ~i=1 I ai - - bi], w e

conclude that no two centers of stars of size /> 2 can receive a common color and in each star of size >~2 the color that occurs in the center x also occurs in N j ( x ) . Therefore, if ~o(x)=ci, x E A (resp. B), ci occurs in B (resp. A) at most

s (resp. t)times, if x is the center of a star of size ~>s (resp. t); l t i m e s , i f x i s t h e center of a star of size l, 2 ~ l < s ; and 1 times, if x is incident with an independent edge.

Furthermore, if ¢i does not occur in A (resp. B), then ci can occur at most s times in B (resp. A). Now (1) is a direct result of the vertex coloring using at most n colors. Conversely, if (1) is true, then the biconformable vertex coloring can be obtained by assigning the colors to the vertices following the processes: (i) if x y is an independent edge of J , then ~p(x)--= q~(y), and for each independent edge one color is used. (ii) All centers of stars of size at least 2 are colored differently. (iii) I f a center x E A (resp. B) of a star of size I is colored with ci, then color the vertices of NG(x) with ci min{s, l} times (resp. min{t, l} times). (iv) I f there are Sl and s2 vertices in A and B respectively which are not colored yet, then use {st/t] and Fs2/s] colors to color them respectively. As explained in the necessity part, the coloring obtained by the above processes is biconformable. []

It is easy to see that if H is a subgraph of a biconformable graph G such that

V ( H ) = V ( G ) and A ( H ) = A(G), then H is also biconformable. But if G is not bi-

conformable, we may still have a subgraph H of G with A ( H ) = A ( G ) and H is biconformable. In what follows, we obtain a necessary and sufficient condition for an equibipartite maximal graph G which contains a subgraph H such that H is not biconformable and A ( H ) = A( G).

Proposition

2.3. L e t G = Kn, n \ E ( J ) , where J = ( m l , . . . , ms; n l , . . . , n t ; r)n. Then G con- tains an equibipartite subgraph H with A ( H ) = A ( G ) which is not biconformable i f and only i f either n<~ml + n l , or (1) is not true.

Proof. Assume that H is an equibipartite subgraph of G such that A ( H ) = A ( G ) which is not biconformable. First, if V ( H ) = V(G), then clearly G is not biconformable ei- ther. By Proposition 2.2, (1) is not true. On the other hand, if V ( H ) ~ V(G), then

f V(H)[

= 2 ( n - 1) and A ( H ) = A ( G ) = n - 1. Let JH = K n - l , n - l \ E ( H ) and u,v be two

vertices in V ( G ) such that H is a maximal subgraph of HI = G \ { u , v}. Furthermore, Let u' and v' be two vertices in V(G) such that degj (u') = m 1, degj (v') = n l, H ' = G \ { u', v' } and JH' = K , - I , n - I \ E ( H ' ) . Now we have

B.-L. Chen et al./Discrete Mathematics 184 (1998) 4960 55 Since H is not biconformable, H is type 2. By Theorem 1.2, z t ( H ) = n + 1 (A ( H ) = n - 1 ) if and only if e(JH) + m(JH ) ~< n -- 2. This implies that 2n - m 1 - n l - 2 ~< n - 2, and therefore n ~<mj + nl.

Conversly, in the case that (1) is not true, then G is not biconformable. Hence the existence o f H is obvious. Assume that n ~< m I + n 1, degc,(xl ) = ml, and degc,(yl ) = n 1. Let H = G \ { x , , y , } and JH = J \ { x , , y , } . Again

e(Jl4 ) + m(JH ) = 2n -- ml -- nl -- 2 <~ n -- 2.

Since H is o f order 2 ( n - l ) and A ( H ) = n - I . By Theorem 1.2, H is a type 2 subgraph o f G, and hence not biconformable.

Corollary 2.4. L e t G : K n , n \ E ( J ) , where J = (ml . . . ms;n1 . . . nt;r)n and G is bicon- Jormable. Then every equibipartite subgraph H with A ( H ) = A ( G ) is biconJormable

i f and only i f n > ml + nl.

Proof. It is a direct result o f Propositions 2.2 and 2.3. []

We note here that, from Corollary 2.4, if we can find a type 2, biconformable graph

G = K n , n \ E ( J ) where J = fml . . . ms; nl . . . . ,nt; r), and n > ml + nl, then the bicon-

formability conjecture can be disproved. Not surprisingly, we shall see that the graph in Fig. 3 is one o f this kind.

3. The problem of distributing colored balls (DCB)

In order to obtain a good necessary condition for a type 1 maximal equibipartite graph with maximal degree n - 1 (hopefully this condition is also sufficient), we in- troduce a problem which is formulated by biconformable total colorings. The details will be explained in next section.

D C B Problem. Suppose that we have t different colored balls and there are ni balls o f the ith color, i = 1 , 2 , . . . , t . Without loss o f generality, let nx >~n2~>... >~nt. The DCB problem is to determine the minimum number o f boxes which are needed to distribute all the balls given

(i) the ith box contains exactly one ball o f the ith color and in total at most ni balls, i = 1,2 . . . t;

(ii) the j t h box contains at most t balls for each j > t ; and (iii) every box consists o f different colored balls.

Let N = ( n l , n 2 , . . . , n t ) and b ( N ) denote the minimum number o f boxes we need to distribute the colored balls properly into different boxes. In order to find b ( N ) we need the Fulkerson's theorem on digraphical sequence.

Theorem 3.1 (Fulkerson [4]). A sequence (s l, t l ),

($2,

t2) . . . (Sp,

tp) of

ordered pairs o f nonnegative integers with si the outdegree, ti the indegree, and sl >~s2>~... >~Sp,

56 B.-L. Chen et al./ Discrete Mathematics 184 (1998) 49~0 Box 1 Box 2 Box 3 Box 4 B o x 5

I w l l l 2 l

[®lll21

(a) Box 1 Box 2 Box 3 Box 4 B o x 5

Fig. 4(a) and (b).

1 )[1/314[o]

[ lII2/

I®lll -I

l 121olol

(b)

is digraphical if and only if

(i) s i < ~ p - 1 and t i < ~ p - 1 for 1 <.i<~p;

(ii) ZP=l si = ~P=I ti;

(iii) Y'~i~l si <~ Zi~=l min{n - 1, ti} + Y~4P=n+, min{n, ti} for 1 <~ n < p.

Before we prove the main lemma, we shall use an example to explain our idea. In Fig. 4(a), we have N = (5, 5, 3, 3) and b ( N ) = 5. The numbers respresent the colors of

the colored balls.

Since there are three boxes in Fig. 4(a) which are not full, we can fill in some dummy balls with color 0 without changing the minimum number o f boxes. Fig. 4(b) is such an adjustment.

Now we can define a digraph G by way o f Fig. 4(b). Let V(G) = {vl, v2, v3, v4, u~, wl,

Wz, W3,W4} where vi represents the box which contains the ith color ball, i = 1,2,3,4,

and u~ represents the extra box in which we can distribute at most 4 distinct colored balls. Finally, let the wi's represent the dummy balls (one for each). The arcs o f G

can be seen in Fig. 5, the indegree o f vi is n i - 1 which represents that except for the ith color ball, there are n i - 1 balls in the box. Furthermore, if the extra box Ul contains an ith colored ball, then (vi, ul) is an arc o f G, and (wk, vi) is an arc o f G

provided that ith box contains a dummy color ball wk. Clearly, G has a digraphical sequence: (4, 4), (4, 4), (2, 2), (2, 2), (0, 4), (1,0), (1,0), (1,0), (1,0). Since the sequence is digraphical, the property (iii) in Theorem 3.1 holds and we shall use (iii) to find

b(N).

Proposition 3.2. Let N = (n l, n2 . . . nt) where n~ >1 n2 >1 • " >~ nt are positive integers. Then

- - Y-~i=k+, min{k, ni I } ] + t.

E k a ni ~ k I min{k, ni} t

b ( N ) = m a x

B.-L. (-;.hen et al./Discrete Mathematics 184 (1998) 4 9 4 0 57 V I U Fig. 5. ~ w W 2 W 3 - . - - . . . ~ W 4

Proof. By the definition of the DCB problem, b(N)>~ t. Let 7 be the minimum number of extra boxes we need, i.e., b(N) = t + 7. Then we can define a digraph G similar

to Fig. 5 with V ( G ) = { v l , v 2 , . . . , v t , Ul,U2 . . . u ; . , W l , W 2 . . . . ,w.;t}, and the degree se-

quence:

(nl -- 1,nl - 1),(n2 -- 1,n2 -- 1) . . . ( n t - - 1 , n t - 1),

(0, t), . . . , (O,t), ( 1 , 0 ) , . . . , ( ! , 0 ) .

y

~; times 7 t times

(The number of dummy balls are decided by the number of extra boxes.) Since the sequence is digraphieal, by Theorem 3.1, we have

k I v(a)l

~ ( n i - 1 ) ~ < ~ m i n { k -

1,ni-

1} + ~ min{k, ti}

i=1 i = l i = k + l

for each 1 <~k<~t where ti is the indegree of a vertex. Hence

ni - k ~ < ~ m i n { k - 1 , h i - 1 } + ~ min{k, t i } + k T ,

58 B.-L. Chen et al./Discrete M a t h e m a t i c s 184 (1998) 4 9 - 6 0

i . e ,

k ) ' > ~ n i - m i n { k - 1 , n i - I} + k -

i = l

This implies that

~>~ ni - ~ m i n { k , ni} -

i = l

for each 1 ~< k ~< t. Thus

t

S min{k, ti}.

i = k + l

I~-]~k ni -- ~-~_ min{k, ni} -- ~'~i=k+l t min{k, ni - 1} ]

b(N)>~ max i=1 ~ , - 1 _{+ t.} ~k~<t k Now let -i - - - - Z i : k + l m i n { k , ni [ y ' = m a x

- ~ : t r t i

~ ] ~ : t m i n { k ' n i } t - 1} l<~k<~t k { "

Conversely, we see that (iii) holds for l <~k<t + 7~t + 7 ~. Therefore b(N)<~y~ + t.

( b ( N ) is a minimum.) And we have the proof. []

Now we are ready for the main theorem.

4. The main results

Let G = K n . , \ E ( J ) where J = { m l , m 2 .... ,ms; nl,n2 . . . nt; r)n. Then J can be decomposed into three edge-disjoint induced subgraphs; H1 is induced by the stars with centers xl,x2 . . . x~, H2 is induced by the stars with centers yl, Y2,..., Yt, a n d / / 3 is in- duced by all the independent edges. (Following the notations in Section 2.) Let ~Pl, (P2 and ~o3 be the vertex colorings o f G restricted on V(Ht ), V(Hz) and V(H3), respec- tively, such that the mages o f ~pf, ~o2 and q~3 are mutually disjoint. Clearly, the union of ~o j, ~02 and q~3 is a vertex coloring o f G using I q~ l ( V(HI ))[ + [~02( V(H2 ))1

+ I q~3

( V(H3 ))1

colors. Now if n > ~ b ( M ) + b ( N ) + r where M = ( m l , m 2 , . . . , m s ) and N = ( n l , n 2 . . . nt), then we can reserve b ( M ) colors for qh, b ( N ) colors for q~2 and r colors for ~o3. A biconformable vertex coloring ~o can be obtained by the following assignment: (i) in V(H1 ), let ~o(xi)= ~i; for the vertices in Nc(xi), at most s vertices can be colored with c~i, i = 1,2 .... ,s, and each occurs at most s times; (ii) in V(H2), let ~o(yi)=fli; for the vertices in NG(yi), at most t vertices can be colored with fli, i = 1,2 . . . t, and each occurs at most t times; and (iii) in V(H3), for each an independent edge xy, let ~o(x)--q~(y) = °/i. Then we have the following lemma.

L e m m a 4.1. Let G = K n , n \ E ( J ) where J = ( m l , m z , . . . , m s ; nl,n2 . . . ms; r)n. Also let

M = (ml, m2 . . . ms) and N = (nl, n2 . . . nt). Then G is biconforrnable provided that n >~b(M) + b ( N ) + r.

B.-L. Chen et al./ Discrete Mathematics 184 (1998) 49~50 59 The above inequality plays an important role in the classification o f maximal equibi- partite graphs with m a x i m u m degree n - 1.

Proposition

4.2. Let G = K , , , , \ E ( J ) where J = (ml,m2 . . . ms; nl,n2 . . . . ,nt, r)n. Let M =(ml,m2 ... ms.) and N =(nl,n2,...,nt). I f G is type 1, then n > j b ( M ) + b ( N ) + r . P r o o f . F o l l o w i n g the notation before L e m m a 4.1, let V ( G ) = V(H1 ) U V(H2)U V(H3). Since G is type 1, G is biconformable. Therefore, for each type 1 total coloring ~o o f G, we need at least b(M) colors to color V(H1 ) and at least b ( N ) + r colors to color the rest vertices o f G. In order to prove the proposition, it suffices to show that no color that occurs in V(HI ) can occur in V(H2)U V(H3) and no color that occurs in V(H2) can occur in V(H1)U V(H3). The p r o o f o f the second statement is similar to the first. Thus we prove the first statement only.

First, we see that in the total coloring ~o each color occurs at most n + 1 times (either on vertices or edges), and in total there are 2n vertices and n 2 - ( 2 n - s - t - r ) edges in G. Furthermore, only the colors occurs on the center o f stars or the vertices o f independent edges can occur n + 1 times, hence there are s + t + r colors which occurs n + 1 times and n for each o f the other colors. Now, if x is one o f the b(M) colors for V(HI ) which does not occur on the centers, then it occurs on the major vertices in V(HI ). As a consequence, x cannot occur on V(H2) U V(H3 ). On the other hand, i f x is a color on a center o f a star in H1, then x cannot be used in coloring V(H2) U V(H3). For otherwise, x occurs only n times in G. This concludes the proof. []

N o w we obtain a class o f maximal equibipartite graphs with m a x i m u m degree n -

1

which are type 2.

P r o p o s i t i o n 4.3. Let d = (ml,m2 . . . ms; nl,n2 .... ,nt; r),. I f nl = n r + l = t ~ > r + l , then G = K n . n \ E ( J ) is type 2.

P r o o f . Let M = ( r n l ) and N = ( t , t . . . t, nr+2,...,nt). Then b ( M ) = m l and b(N)>~t. This implies that b(M) + b(N) + r>~ml + t + r. By Proposition 4.2, i f b ( N ) > t , then b ( M ) + b ( N ) + r > n and G is type 2. W e are done. Thus assume that b ( N ) = t and G is type 1. Let ~o be a type 1 total coloring o f G and we use the colors cl, c2 . . . cn. Let the centers o f stars (in J ) o f size at least 2 be x l ; y l , y 2 ... Yr. Since ~0 is bicon- formable, the colors that occur on Nj(xl ) are all distinct such that there is one vertex colored with ~O(Xl), and for 1 <~i<<.r + 1, the color that occurs on Nj(yi) is exactly the same as ~o(yi). (Note that by L e m m a 4.1, V(HI), V(H2), and V(H3) use different colors.) Without loss o f generality, let the colors that occur on Nj(xl ) be Cl, c2 . . . era,

I l r + l

and the colors that occur on k.)i=l N ( y i ) be C,n,+j,Cm,+2 . . . . ,C,,,+~+1. NOW in B, ex- cept for Yi, Y2 . . . Yt, there are n - t major vertices which are o f degree n - 1 and C,n, ~i, does not occur on these vertices for each 1 ~< i ~<r + 1. Therefore, there exists a matching Ti which is incident with these vertices and each edge o f Ti is colored with Cm,+i. Since Cm,+i occurs on Nj(yi), T, is incident with xl, i.e., there exists an edge

60 B.-L. Chen et al./ Discrete Mathematics 184 (1998) 49~60

xlyt+j,, l<~ji<<,r, which is colored with Cm,+i. Now, in total, we have r + 1 edges o f the form Xl Yt+j, which have distinct colors. This is not possible. Thus G must be type 2.

As a special case o f the graphs in Proposition 4.3, let G = K ~ , n \ E ( J ) where J = (ml; t,t . . . . ,t; r),, l <<.r < t and n = t 2 + r + 1. Then G is type 2. Furthermore, by Proposition 2.4, since 1 + ( m l - - 1 ) + t = r = ml + t = r = n, G is biconformable. Also, n > m t + t, every equibipartite subgraph H with A ( G ) = A ( H ) is biconformable. Thus we have the following result which shows that the biconformability conjecture is not true in general. []

Proposition

4.4. L e t G = K n , n \ E ( J ) where J = (ml; t,t . . . t; r)n, l ~ r < t and n =

t 2 + r ÷ 1. Then G is a counterexample to Conjecture 1.5.

Note that, by Propositions 4.2 and 4.4, the condition b ( M ) + b ( N ) + r<~n for a type 1 graph is necessary but not sufficient. On the other hand, the biconformability conjecture is true for the case when A ( G ) = n as mentioned in L e m m a 1.4. Hence the counterexample obtained here is sharp with respect to A(G).

With the work we have done so far we believe the following Conjecture might be true.

Conjecture

4.5. Let G =Kn, n \ E ( d ) where J = (ml,m2 .... ,ms; nl,n2 .... ,nt; r)n. Then

G is type 2 i f and only i f either

n < b ( m l , m 2 , . . . , m s ) + b ( n l , n 2 .... , n t ) + r , o r s = l a n d n l = n r + l = t > ~ r + l .

Acknowledgements

The authors would like to express their thanks to the referees for their helpful comments.

References

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