Cycle Lengths in Expanding Graphs

Abstract

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.