Cycle lengths modulo $k$ in expanders

TL;DR

This paper proves that large alpha-expanding graphs contain cycles of all lengths modulo k when alpha exceeds a certain threshold related to the smallest prime divisor of k, addressing a question in graph theory.

Contribution

The paper establishes a near-optimal condition on alpha-expansion for the existence of cycles of all lengths modulo k in large graphs, answering a question by Friedman and Krivelevich.

Findings

Large alpha-expanding graphs contain cycles of all lengths modulo k for alpha > 1/(p-1).

The result is nearly tight, with counterexamples for smaller alpha.

The minimal prime divisor p of k determines the threshold for cycle length existence.

Abstract

Given a constant $\alpha>0$, an $n$-vertex graph is called an $\alpha$-expander if every set $X$ of at most $n/2$ vertices in $G$ has an external neighborhood of size at least $\alpha|X|$. 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 $\alpha>\frac{1}{p-1}$ every sufficiently large $\alpha$-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 $\alpha<\frac{c}{p-1}$, there exist arbitrarily large $\alpha$-expanding graphs with no cycles of length $r$ modulo $k$, for some $r \in \{0,\ldots,k-1\}$.