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By B. Bollobás (Eds.)

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By a theorem of Vizing the chromatic index of the graph G is either A or A + 1, If G is a regular graph of degree A, then q ( G )= A if and only if the edges of G can be partitioned into l-factors. A partitioning of the edges of G into l-factors is called a l-factorization of G. The graphs G,, GZ, and G, being the complete graphs K , , K,, and K , respectively, have l-factorizations. The graph G, is the 6-partite graph K4,4,4,4,4,4, and a l-factorization is readily found. One is naturally lead to conjecture that the graph G, has a l-factorization for each n 2 2.

Bollobas, P. Erdos, M. Simonouits, E. Szemerkdi where a = /1 = rIJs s + , V , j = ( s + l ) m . To obtain an upper bound of d in terms of a, we apply Lemma 5 to the bipartite graph determined by the classes Uzs,ss+lV , (=first class) and V, (=second class). We find that G" I K z ( r , t ) with t = ( 1-o(l))drnra-(r-l1. (7) By the assumption G " 3Kz(r, 22'C'c;n) and by (7) (8) drnrC1a-(rCIJ< ( 1+0(1))2~'-'c;. Let us assume that d > 2c, (this will be shown later). From (8) and c;

Using the first part we may assume that the edges are regularly distributed. We may repeat part (A) of the proof of Theorem 7 replacing e( ) by h( ) and c by c'. Then we may assume that G' = G", but have to decrease c': replace the original condition by condition h ( G " ) s c"n2. How we define the graph R k as in the beginning of (B) of the proof of Theorem 7. (B) We prove that if n is sufficiently large and R k 3 CJ,then G" 3 CJ(r),where r is fixed, but arbitrarily large. Exactly as in the proof of Theorem 7, we can prove that G " contains at least c , n J cycles CJ,where c , > O is a constant.

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