Hitting time for Hamilton cycles in pseudorandom graphs

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 $(n,d,λ)$-graph with $d/λ\ge C$, 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 $(n,d,λ)$-graphs with $d/λ\ge C$ 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.

Theorems & Definitions (14)

• Theorem 1.1: Draganic-et-al
• Theorem 1.2
• Corollary 1.3
• Corollary 1.4
• Theorem 1.5
• Theorem 1.6: Draganic-et-al
• Theorem 1.7
• Lemma 2.1: janson-book
• Lemma 2.2: alon-chung
• Proposition 2.3