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 n₀, C and A = O(1/α) such that for every α-expander G on n≥n₀ vertices and every integer $$\ell \in \left[ {C\log n,\tfrac{n}{C}} \right]$$, G contains a cycle whose length is between $$\ell$$ and $$\ell$$+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 $$\Omega \left({\tfrac{{{\alpha ^3}}}{{\log (1/\alpha )}}} \right)$$ 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 $$\beta < 1/20$$ there exist positive constants $${b_1} = O\left({\tfrac{1}{{\log (1 - \beta )}}} \right)$$ and b₂ = O(β) such that every β-graph G on n vertices contains a cycle of length $$\ell$$ for every integer $$\ell \in \left[ {{b_1}\log n,\left({1 - {b_2}} \right)n} \right]$$; the order of dependence of b₁ and b₂ on β is optimal.