Resilient degree sequences with respect to Hamilton cycles and matchings in random graphs

TL;DR

This paper extends classical Hamilton cycle theorems to random graphs, establishing resilience conditions on degree sequences that guarantee Hamiltonicity and perfect matchings, with near-optimal bounds and new conjectures.

Contribution

It proves a resilience version of Pósa's theorem for Hamilton cycles in random graphs and explores the limitations of Chvátal's theorem, proposing a conjecture for perfect matchings.

Findings

Resilience version of Pósa's theorem established for random graphs.

Degree sequence conditions ensure Hamiltonicity with near-optimal bounds.

A conjecture is formulated for resilience conditions guaranteeing perfect matchings.

Abstract

P\'osa's theorem states that any graph $G$ whose degree sequence $d_1 \le \ldots \le d_n$ satisfies $d_i \ge i+1$ for all $i < n/2$ has a Hamilton cycle. This degree condition is best possible. We show that a similar result holds for suitable subgraphs $G$ of random graphs, i.e. we prove a `resilience version' of P\'osa's theorem: if $pn \ge C \log n$ and the $i$-th vertex degree (ordered increasingly) of $G \subseteq G_{n,p}$ is at least $(i+o(n))p$ for all $i<n/2$, then $G$ has a Hamilton cycle. This is essentially best possible and strengthens a resilience version of Dirac's theorem obtained by Lee and Sudakov. Chv\'atal's theorem generalises P\'osa's theorem and characterises all degree sequences which ensure the existence of a Hamilton cycle. We show that a natural guess for a resilience version of Chv\'atal's theorem fails to be true. We formulate a conjecture which would repair this guess, and show that the corresponding degree conditions ensure the existence of a perfect matching in any subgraph of $G_{n,p}$ which satisfies these conditions. This provides an asymptotic characterisation of all degree sequences which resiliently guarantee the existence of a perfect matching.