On the Distribution of Cycle Lengths

A. Gyarfas COMPUTER AND AUTOMATION INSTITUTE OF THE HUNGARIAN ACADEMY OF SCIENCES BUDAPEST_ HUNGARY J. Koml6s and E. Szemeredi MATHEMATICAL INSTITUTE OF THE HUNGARIAN ACADEMY OF SCIENCES BUDAPEST_ HUNGARY The set of different cycle lengths of a graph G is denoted by C(G). We study how the distribution of C(G) depends on the minimum degree of G. We prove two results indicating that C(G) is dense in some sense. These results lead to the solution of a conjecture of Erdos and Hajnal stating that for suitable positive constants a, b the following holds: 2: i7 ~ a log[o(G)], iEC(G) where o(G) denotes the minimum degree of G.