Graphs with k odd cycle lengths

Gyarfas, A., Graphs with k odd cycle lengths, Discrete Mathematics 103 (1992) 41-48. 41 If G is a graph with k ~ 1 odd cycle lengths then each block of G is either K2k+Z or contains a vertex of degree at most 2k. As a consequence, the chromatic number of G is at most 2k + 2. For a graph G let L(G) denote the set of odd cycle lengths of G, i.e., L( G) = {2i + 1: G contains a cycle of length 2i + 1}. With this notation, bipartite graphs are the graphs with IL(G)I = 0. Bollobas and Erdos asked how large can the chromatic number of G be if IL(G)I = k. They conjectured that IL(G)I = k implies x(G) ~ 2k + 2 and this is best possible considering G = K2k+2· The case k = 1 is checked by Bollobas and Shelah (see [1, p. 472] for the motivation). Gallai suspected that a stronger statement is true, namely if G is 2-connected, IL(G)I = k, G =I= K 2k+2 then the minimum degree of G is at most 2k. The aim of this paper is to prove this stronger version of the original conjecture. Theorem 1. If G is a 2-connected graph with minimum degree at least 2k + 1 then IL(G)I = k ~ 1 implies G = K2k+2· Assuming that IL(G)I = k, Theorem 1 clearly allows to color the vertices of the blocks of G with at most 2k + 1 colors except when a block is a K 2k+ 2 • Thus the following corollary is obtained. Corollary. If IL(G)I = k ~.1 then the chromatic number of G is at most 2k + 1, unless some block of G is a K2k+2 . (If there is such a block, then the chromatic number of G is 2k + 2.) 0012-365X/92/$05.00 © 1992-Elsevier Science Publishers B.V. All rights reserved