Hamiltonicity of expanders: optimal bounds and applications

TL;DR

This paper proves that certain expanders are guaranteed to contain Hamilton cycles, resolving a longstanding conjecture and advancing understanding of Hamiltonicity in sparse graphs with broad applications.

Contribution

It establishes that all sufficiently strong expanders are Hamiltonian, confirming a conjecture and extending results to random Cayley graphs.

Findings

Every $C$-expander is Hamiltonian for some constant $C>0$

Confirms the Krivelevich and Sudakov conjecture from 2003

Implications for Hamiltonicity in random Cayley graphs

Abstract

An $n$-vertex graph $G$ is a $C$-expander if $|N(X)|\geq C|X|$ for every $X\subseteq V(G)$ with $|X|< n/2C$ and there is an edge between every two disjoint sets of at least $n/2C$ vertices. We show that there is some constant $C>0$ for which every $C$-expander is Hamiltonian. In particular, this implies the well known conjecture of Krivelevich and Sudakov from 2003 on Hamilton cycles in $(n,d,\lambda)$-graphs. This completes a long line of research on the Hamiltonicity of sparse graphs, and has many applications, including to the Hamiltonicity of random Cayley graphs.