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 nn0 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.

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Acknowledgements

The authors are grateful to Rajko Nenadov for his contribution to this paper. The authors also wish to thank Asaf Cohen, Wojciech Samotij and Leonid Vishnevsky for their input and remarks.

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Correspondence to Michael Krivelevich.

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Research supported in part by USA-Israel BSF grant 2018267, and by ISF grant 1261/17.

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Friedman, L., Krivelevich, M. Cycle Lengths in Expanding Graphs. Combinatorica 41, 53–74 (2021). https://doi.org/10.1007/s00493-020-4434-0

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Mathematics Subject Classification (2020)

  • 05C38
  • 05C48
  • 05C35