Cycle Lengths in Expanding Graphs

For a positive constant alpha a graph G on n vertices is called an alpha-expander if every vertex set U of size at most n/2 has an external neighborhood whose size is at least alpha|U|. We study cycle lengths in expanding graphs. We first prove that cycle lengths in alpha-expanders are well distributed. Specifically, we show that for every 0 < alpha <= 1 there exist positive constants n(0), C and A = O(1/alpha) such that for every alpha-expander G on n >= n(0) vertices and every integer , l is an element of[C log n,n/c] G contains a cycle whose length is between l and l+A; the order of dependence of the additive error term A on alpha is optimal. Secondly, we show that every alpha-expander on n vertices contains Omega(alpha(3)/log(1/alpha)) n different cycle lengths. Finally, we introduce another expansion-type property, guaranteeing the existence of a linearly long interval in the set of cycle lengths. For beta > 0 a graph G on n vertices is called a beta-graph if every pair of disjoint sets of size at least beta n are connected by an edge. We prove that for every there exist positive constants b(1)-O(1/log(1/beta))and b(2) = O(beta) such that every beta-graph G on n vertices contains a cycle of length l for every integer l is an element of[b(1) logn,(1-b(2))n]; the order of dependence of b(1) and b(2) on beta is optimal.