Cycle lengths modulo <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" id="d1e22" altimg="si21.svg"><mml:mi>k</mml:mi></mml:math> in expanders

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