Hitting time for Hamilton cycles in pseudorandom graphs

TL;DR

This paper proves that in certain pseudorandom graphs, the time to first Hamilton cycle during a random edge process matches the time when the minimum degree reaches two, confirming longstanding conjectures.

Contribution

It establishes the hitting time for Hamilton cycles in $(n,d,)$-graphs with large degree-to-eigenvalue ratio, resolving open questions and determining sharp thresholds.

Findings

Hitting time for Hamilton cycles coincides with minimum degree 2

Sharp threshold for Hamilton cycles in pseudorandom graphs

Extension to multiple edge-disjoint Hamilton cycles

Abstract

Consider the random subgraph process on a base graph $G$ with $n$ vertices: we generate a sequence $\{G_t\}_{t=0}^{|E(G)|}$ by taking a uniformly random ordering of the edges of $G$ and then adding these edges one by one to the empty graph $G_0$ on the same vertex set. We prove that there is a constant $C > 0$ such that if $G$ is an $(n,d,\lambda)$-graph with $d/\lambda \ge C$, then with high probability, the hitting time for the appearance of a Hamilton cycle coincides with the hitting time for reaching minimum degree $2$. This resolves questions posed by Alon--Krivelevich in 2019 and by Frieze--Krivelevich in 2002. As a consequence, we determine the sharp threshold for Hamilton cycles in $(n,d,\lambda)$-graphs with $d/\lambda\ge C$ for all $d$ sufficiently large. Lastly, we extend our result to the minimum degree $2k$ versus $k$ edge-disjoint Hamilton cycles setting for $k \leq c\cdot \min\{d,\log n\}$ where $c$ is a constant depending on $C$. This advances on a question asked by Frieze.