Cycle lengths modulo [formula omitted] in expanders

Abstract

Given a constant α>0, an n-vertex graph is called an α-expander if every set X of at most n/2 vertices in G has an external neighborhood of size at least α|X|. Addressing a question posed by Friedman and Krivelevich in [Combinatorica, 41(1), (2021), pp. 53–74], we prove the following result:Let k>1 be an integer with smallest prime divisor p. Then for α>1p−1 every sufficiently large α-expanding graph contains cycles of length congruent to any given residue modulo k.This result is almost best possible, in the following sense: There exists an absolute constant c>0 such that for every integer k with smallest prime divisor p and for every positive α<cp−1, there exist arbitrarily large α-expanding graphs with no cycles of length r modulo k, for some r∈{0,…,k−1}.