Cycle Lengths in Expanding Graphs

For a positive constant α a graph G on n vertices is called an α-expander if every vertex set U of size at most n /2 has an external neighborhood whose size is at least α| U |. We study cycle lengths in expanding graphs. We first prove that cycle lengths in α-expanders are well distributed. Specifically, we show that for every 0 < α ≤ 1 there exist positive constants n 0 , C and A = O (1 / α) such that for every α-expander G on n ≥ n 0 vertices and every integer , G contains a cycle whose length is between and + A ; the order of dependence of the additive error term A on α is optimal. Secondly, we show that every α-expander on n vertices contains 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 β > 0 a graph G on n vertices is called a β -graph if every pair of disjoint sets of size at least βn are connected by an edge. We prove that for every there exist positive constants and b 2 = O(β) such that every β -graph G on n vertices contains a cycle of length for every integer ; the order of dependence of b 1 and b 2 on β is optimal.