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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 $α\left|U\right|$. 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 $0<α\leq1$ there exist positive constants $n_{0}$, $C$ and $A=O(1/α)$ such that for every $α$-expander $G$ on $n\geq n_{0}$ vertices and every integer $\ell\in\left[C\log n,\frac{n}{C}\right]$, $G$ 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 $Ω\left(\frac{α^{3}}{\log\left(1/α\right)}\right)n$ 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 $β>0$ 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 $β<1/20$ there exist positive constants $b_{1}=O\left(\frac{1}{\log\left(1/β\right)}\right)$ and $b_{2}=O\left(β\right)$ such that every $β$-graph $G$ on $n$ vertices contains a cycle of length $\ell$ for every integer $\ell\in\left[b_{1}\log n,(1-b_{2})n\right]$; the order of dependence of $b_{1}$ and $b_{2}$ on $β$ is optimal.