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 \(\ell \in \left[ {C\,\log \,n,\,\frac{n}{C}} \right]\), G contains a cycle whose length is between l and l+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 \(\Omega \left( {\frac{{{\alpha ^3}}}{{\log \left( {1/\alpha } \right)}}} \right)n\) 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 \({b_1} = O\left( {\frac{1}{{\log \left( {1/\beta } \right)}}} \right)\) and b₂ = O(β) such that every β-graph G on n vertices contains a cycle of length l for every integer l = ∈ b₁ logn, (1 - b₂)n]; the order of dependence of b₁ and b₂ on β is optimal.

对于正常数α，如果每个大小最大为n / 2的顶点集U具有一个大小至少为α|的外部邻域，则n个顶点上的图G称为α展开器。U |。我们在扩展图中研究周期长度。我们首先证明α膨胀机中的周期长度分布良好。具体而言，我们表明，对于每个0 <α ⪯ 1存在正的常数Ñ ₀，c ^和A = 0（1 / α），使得对于每个α-膨胀ģ上n≽n ₀的顶点和每一个整数\（\ ell \ in \ left [{C \，\ log \，n，\，\ frac {n} {C}} \ right] \），G包含一个长度在l到l + A之间的循环；加性误差项A对α的依赖顺序是最佳的。其次，我们证明n个顶点上的每个α展开器都包含\（\ Omega \ left（{\ frac {{{\\ alpha ^ 3}}} {{\ log \ left（{1 / \ alpha} \ right）} }} \ right）n \）不同的循环长度。最后，我们引入了另一个扩展类型属性，以保证在循环长度集中存在线性长间隔。对于β> 0的曲线图ģ上Ñ顶点被称为β -图表如果每对不相交的集合大小中的至少βn通过边连接。我们证明对于每个β } <1/20，都有正常数 \（{b_1} = O \ left（{\ frac {1} {{\ log \ left（{1 / \ beta} \ right）}}}} \右）\）和b ₂ = O（β），使得每个β-图ģ上ñ顶点包含长度的周期升每整数升=∈ b _1个日志ñ，（1 - b ₂）ñ ]; b ₁和b ₂对β的依存顺序是最佳的。