THE ACHROMATIC INDEXES OF THE REGULAR COMPLETE MULTIPARTITE GRAPHS

DISCRETE MATHEMATICS ELSEVIER Discrete Mathematics 141 (1995) 61 -66

The achromatic indices of the regular complete multipartite

graphs*

N a m - P o C h i a n g x, H u n g - L i n F u *

Department o[Applied Mathematics, National Chiao Tung Unil~ersitv. Hsin-Chu, Taiwan, Republic ~!/Uhim~

Received 14 April 1991; revised 31 August 1993

Abstract

In this paper, we study the achromatic indices of the regular complete multipartite graphs and obtain the following results:

(1) A good upper bound for the achromatic index of the regular complete multipartite graph which gives the exact values of an infinite family of graphs and solves a problem posed by Bouchet.

(2) An improved Bouchet coloring which gives the achromatic indices of another infinite family of regular complete multipartite graphs.

1. Introduction

An edge k-coloring of a simple graph G = ( V ( G ) , E(G)) is a surjection from E(G) to the set { 1, 2 ... k } (which represents colors) so that any two incident edges of G receive different colors. Moreover, if for each pair of colors Cl and c2 there are incident edges el and e2 so that ei is colored cl, then the coloring is complete. The largest k so that there exists a complete edge k-coloring of G is the achromatic index ~P'(G) of G. The basic concepts related to graph colorings can be referred to [1, 3, 4, 8, 9].

The achromatic index of a complete graph, a regular complete multipartite graph, has been studied by Bouchet et al. in [2, 7, 11], respectively. Mainly, partial results are obtained. In particular, on regular complete multipartite graphs, Bouchet proved the following theorem.

Theorem 1.1. Suppose that q is an odd integer and equal to the order of a projective plane. I f n and m are integers such that n l q + 1 and m =q(q + 1)/n, then tP'(K,(,,I)>~ '~ Research supported by the National Science Council of the Republic of China (NSC79-0208-M009-331. x Present address: Department of Applied Mathematics, Tatung Institute of Technology, Taipei, Taiwan, Republic of China.

* Corresponding author.

0012-365X/95/$09.50 © 1995 Elsevier Science B.V. All rights reserved

62

N.-P. Chiang, H.-L. Fu / Discrete Mathematics 141 (1995) 6 1 4 6

q(n-- 1)m where K.t,, ] denotes the regular complete n-partite graph with each partite set consisting of m vertices.

Furthermore, Bouchet posed a problem asking whether the above inequality could actually be an equality. In this paper, we find an upper bound for the achromatic index of the regular complete multipartite graph which forces the inequality to be an equality and a complete edge coloring for some kind of regular complete multipartite graphs which gives the achromatic indices of another class of infinitely m a n y graphs.

2. The upper bound

In this section, we mainly give an upper bound for 7J'(K.[m]). We start with some definitions.

Let g ( x , y , z ) = z ( 2 y ( x - 1 ) - z - - 1) and h ( x , y , z ) = x ( x - 1)yZ/2z. Define fit(re, n) to be the m a x i m u m of g(n, m, t) + 1 and [_ h(n, m, t + 1)J. Moreover, let B(n, m) = min {/~,(m, n): t = 1,2 . . . m ( n - 1 ) - 1 } . Then we have the following lemma.

Lemma 2.1. ~'(K.[,.]) ~<

B(n, m).

Proof. F o r a fixed t<~m(n-1)-1, first suppose that there is a color class F which contains s ~< t edges. Let the n u m b e r of end-vertices of these edges which come from the ith partite set be si. Then it is not difficult to check that the n u m b e r of edges which are incident with the edges of F is 2sm(n-

1)-- 2(~i=11

n S i ( S l . ~ _ . . . _ ~ S i _ l . ~ _ S i + l

+ ... + s . ) ) - s = 2 s m ( n - 1 ) - 2 s 2 +½ ~ = 1 s 2 - s ~ 2 s m ( n - 1 ) - s 2 - s . Thus, the num- ber of color classes is at most 2 s m ( n - 1 ) - s 2 - s + l which is g(n,m,s)+l. Since g(n, m, x) is increasing on [1, re(n-1)-½] when considering n and m as constants, it follows that in this case, 7J'(K.t,,])~g(n,m, t)+ 1.

Secondly, if each color class has more than t edges, then the n u m b e r of color classes is obviously less than or equal to (n-1)m2n/2(t+l), whence ~U'(K.[,,])~< Lh(n, m, t + 1) J in this case.

Combining the above two cases, ~'(K.t,,])<~max{g(n,m, t)+ 1,Lh(n,m, t+ 1)J} =fit for each fixed t. This implies that 7J'(K.[,q)<~B(n,m). []

Hence, if we can find B(n, m) explicitly, then we obtain an explicit upper bound of 7"(K,[m]). So far, we have no answer for the general form. But we do have a very nice

result for special values of n and m. Let

N,-P. Chiang, H.-L. Fu / Discrete Mathematics 141 (1995) 61~56 63

and

Q(x, y, z) = x ( x - 1)y 2 - 4 ( x - 1)z 2y + 2(z 3 -t- 2 2 -- 2). (2} It is easy to see that P(x,y,z)>~O if and only if L h ( x , y , z + l ) J > ~ g ( x , y , z ) + l , and Q(x,y,z)>~O if and only if L h ( x , y , z ) ] > ~ g ( x , y , z ) + l . Consider (1) and (2) as poly- nomials of y, then they are quadratic. Let D 1 and D2 be the discriminants o f ( l ) and (2), respectively. By a direct calculation, Dt and D2 are positive provided that z ~> 2 or : = 1

and n>~3. F u r t h e r m o r e , let 7,6 a n d e be the larger roots of P ( n , y , t ) = O , Q ( n , y , t ) = O and Q(n,y, t + 1 ) = 0 , respectively, and let 7' and e' be the smaller roots of P(n,y, t ) = 0 and Q (n, y, t + 1) = 0, respectively. Also, by solving equations, we have 7' < i~ < 7 < c and c ' < 7. With the above observation, we have the following theorem.

Theorem 2.2. / f t ~> 2 o r t = 1 a n d n >~ 3, then

(i) ~P'(K,[ml)<~g(n,m,t)+ 1 ~ ' m ~ [ 6 , 7 ] and m is an integer; and (ii) ~U'(K, I,,l) ~< L h (n, m, t + 1) l !f m E [ 7, ~] and m is an integer.

Proof. (i) Since m ~ [ 6 , 7 ] , P(n,m,t)<~O and Q(n,m,t)>~O, thus f l , ( m , n ) = g ( n , m , t ) + l . N o w if u is an integer such that t ~ u < ~ m ( n - 1 ) - l , then fl,(n,m)>~g(n,m,u)+l >~ g ( n , m , t ) + 1 =flt(n,m). O n the other situation, if s is an integer such that s < t , then since Q(n, m, t) >>, O, i.e., h(n, m, t) >i g(n, m, t) + 1, hence fls(n, m) >~ h(n, m, s + 1) ~> h(n, m, t) >~ g(n, m, t) + 1 = fit(n, m). This concludes that B(n, m) = fit(n, m) = g(n, m, t) + 1. By L e m m a 2.1, we have p r o v e d (i). Similarly, we can show that in (ii), B(n, m)=[_h(n, m, t + 1 ) J . The p r o o f follows. []

With T h e o r e m 2.2, we are able to obtain B(n, m) for some special n and m.

Corollary 2.3. Let k be an odd integer >13 such that n Jk + 1 and m = k(k + l)/n. Then B ( n , m ) = ( n - l)km. Proof. Let k - 1 ( k + 1 ) ( k - 1) 2 t ( t + 1) t = then m > 2 2n n and P(n, m, t) > 0. Also, ( k + l ) 2 2 ( t + 1) 2 m > - - - 2n n a n d Q ( n , m , t + l ) < O . Thus, m6[y,~]. By T h e o r e m 2.2, B ( n , m ) = L h ( n , m , t + 1 ) J - - L ( n - 1 ) k m J = ( n - 1)km if t ~> 2, or t = 1 and n = 3. Hence, the case left is k = 3 and n = 2. Since m = 6, fli(2, 6) can be obtained directly, i = 1, 2 . . . 5, and B(2, 6) = 18 follows easily.

64 N.-P. Chiang, H.-L. Fu/ Discrete Mathematics 141 (1995) 61-66 Corollary 2.4. For a f i x e d t >I-2, we have

(i) ~ ' ( K 2 [ m ] ) < ~ t ( 2 m - t - - 1)+ 1 / f L 2 t 2 - t/2 ]<~m<~ 2tE + at - 17; and (ii) ~'(Kz[m])<~m2/(t+ 1) /f[-2t 2 +~2tT<~m<~L2(t+ 1)2--½(t+ 1)-- 1J.

N o w by Lemma 2.1, Corollary 2.3 and Theorem 1.1, we obtain our main theorem. Theorem 2.5. Let q be an odd order o f a projective plane. I f n and m are positive integers such that n l q + 1 and m = q ( q + 1)/n, then tP'(K, t m ] ) = q ( n - 1)m.

3. The complete edge coloring

In this section, we use the property of finite projective plane and the construction technique of design theory I-5, 10] to construct a complete edge colorings of a class of infinitely many regular complete partite graphs. Then, we obtain achromatic indices of more graphs.

In the edge coloring, the concept overfull is important. We say that a graph G is overfull if ] V(G)] is odd and [E(G)[ is greater than ½A(G)(] V(G)]-1). Hoffman and Rodger [-6] proved the following theorem.

Theorem 3.1. Let G be a complete partite graph. Then { ~ ( G ) i f G is not overfull; and Z'(G)= (G)+ 1 otherwise.

By the definition of overfull and Theorem 3.1, we know that every regular complete partite graph G of even order has a complete edge A(G)-coloring.

Theorem 3.2. Let q be an order o f a projective plane. Suppose q + 1 = n s + r where O<~r < n and s<~ l.

(i) I f q is odd, then

(a) /f r = 0, then f o r each l such that 0 < ~ l < ~ q - s - n + 2 and In is even, we have 7J'(K,[m])~>(n--1)mq where m = ( s + l)q; and

(b) /fr :/:0, then f o r each I such that 1 <<.l<<.q-s-n+ 2 and n l - r is even, we have ~'(K,[m] ) >~ (n-- 1)mq where m = (s + 1)q.

(ii) I f q is even, then

(a) /f r = 0 , then f o r each l such that 1 <~ I < ~ q - s - n + 2 and In is odd, we have ~'(K,[,,])~>(n-1)mq where m = ( s + l ) q ; and

(b) /fr ¢-0, then f o r each l such that 1 <<.l<<.q-s-n+ 2 and n l - r is odd, we have 7 j' (K, [ml) >/(n -- 1 ) mq where m = (s + l ) q.

Proof. Let (P, p) be a projective plane of order q where P is the set of points and p is the collection of lines in the plane. Then I p l = q 2 + q + 1 and let oo be a point called

N.-P. Chiang, H.-L. Fu/ Discrete Mathematics 14l (1995; 61 4i6 6 5

infinity. Let P ' = P \ {oo} and p' be obtained by replacing all the lines which contain oo with the lines with q points left. It is not difficult to see that (P',p') is a PBD with block sizes q and q + 1. F o r convenience, let the collection of q + 1 lines with q points be H and the collection of qZ lines which contain q + 1 points be V. Since q+ 1 >~l+s+n--1 and q + l = n s + r , we can distribute the lines in H into n parts H1, H2 ... H, such that

(1) there are l + s lines in H1; and

(2) Hi contains at least one line and at most l + s lines, 2 ~<i~< n.

If Hi contains less than l + s lines, then add extra copies of a line in Hi in order that each part Hi, i = 1,2 ... n, contains l + s lines as a result. Let Hj be the set of all points which belong to some line in Hi, i = 1,2 ... n. N o w construct a regular complete n partite graph K,~,. 1, m = ( / + s)q, by using H~(i = 1,2 ... n) as partite set and defining wveE(K,E,.~) if and only if w and v belong to different partite sets.

For each line L in V, let HiL={vsHi: v s L or v is copied from w~Hi and w~L I, i = 1,2 ... n. Since each line in Vintersects each line in H in exactly one point, we have IH/'L=l+s for every i = 1 , 2 ... n. Let V L = H ~ w H ~ u . . . u H L , . Then V c induces

L L L

Knts+ ll

a regular complete n-partite subgraph K,E~+~1=[V ]K.t,~- Each is of order (s + l)n with each vertex of degree ( n - 1)(/+ s). In each case of (it and (ii), is + lln is even. Hence, by T h e o r e m 3.1, there is a complete edge ( n - 1)(/+s)-coloring for each

L L

K,E~+ n and each vertex in K,t~+ll is incident with every color of these ( n - 1 ) ( l + s ) colors.

It is easy to check that for any pair of points from different partite sets appears at

L

least once in K.L~+,i for some L ~ V and the total n u m b e r of edges of these q2 subgraphs is equal to the number of edges in K,t,. j. Hence, these

q2

induced subgraphs form an edge decomposition of K,E,, I. If we color each of these q2 subgraphs with a distinct set of ( n - 1 ) ( / + s ) colors, then we get an edge q a ( n - l ) ( / + s ) - c o l o r i n g of K.E,, j. Since any pair of these qZ lines has at least one point in common, the colorings of all the subgraphs form a complete edge coloring of K,t,,~ using q Z ( s + l t ( n - l ) - - ( n - 1)mq colors. Hence 7~'(K.t,,l)>>-(n-- 1)mq.

By T h e o r e m 3.2 and T h e o r e m 2.2, we obtain the following result.

Theorem 3.3. Let q be an order of a projective plane. Suppose q + 1 =ns + r where

O < r < n and s>~ 1. Let m = ( s + 1)q. Then

(i) l f q is odd and n - r is even, then ~'(K.Eml)=(n--1)mq; and (ii) I f q is even and n - r is odd, then ~P'(K,L,,fl=(n--1)mq.

Proof. By Theorem 3.2, it is clear that tP'(K,~mfl>~(n- 1)mq.

On the other hand, take t = ( s + 1 ) n / 2 - 1 . Then, it is easy to check that 2 ( t + 1)t/ n<m; 2 ( t + 1)2/n<m; P(n, rn, t)>O and Q(n, mt + 1)<0. Hence, m = ( s + 1)q~[7,~] and then B(n, m)= (n - 1)m and tP'(K.Eml) <~(n -- 1)mq. Therefore, tp'(K, M ) -- (n - l )mq. []

66 N.-P. Chiang, H.-L. Fu/ Discrete Mathematics 141 (1995) 6 1 4 6

It c a n be seen f r o m t h e p a p e r s a b o u t a c h r o m a t i c i n d e x t h a t to d e t e r m i n e t h e exact v a l u e of 7~'(G) is v e r y difficult. W e e x p e c t to o b t a i n m o r e r e s u l t s o n this t o p i c in the future.

Acknowledgements

W e w o u l d like to e x p r e s s o u r t h a n k s to D r . S o n g - T y a n g L i u for his h e l p in c h e c k i n g s o m e of t h e r e s u l t s b y u s i n g c o m p u t e r a n d also to t h e referees for t h e i r helpful c o m m e n t s .

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