Local resilience and Hamiltonicity Maker-Breaker games in random-regular graphs

TL;DR

This paper studies the local resilience of random and pseudo-random regular graphs concerning Hamiltonicity and connectivity, showing high resilience and Maker's winning strategies in random regular graphs.

Contribution

It extends local resilience analysis to regular graphs, providing bounds for Hamiltonicity and connectivity, and demonstrates Maker's winning strategies in random regular graphs.

Findings

High probability local resilience of random d-regular graphs to Hamiltonicity

Resilience bounds for Erdős–Rényi graphs with respect to Hamiltonian properties

Maker's winning strategy in Hamilton cycle games on large random regular graphs

Abstract

For an increasing monotone graph property $\mP$ the \emph{local resilience} of a graph $G$ with respect to $\mP$ is the minimal $r$ for which there exists of a subgraph $H\subseteq G$ with all degrees at most $r$ such that the removal of the edges of $H$ from $G$ creates a graph that does not possesses $\mP$. This notion, which was implicitly studied for some ad-hoc properties, was recently treated in a more systematic way in a paper by Sudakov and Vu. Most research conducted with respect to this distance notion focused on the Binomial random graph model $\GNP$ and some families of pseudo-random graphs with respect to several graph properties such as containing a perfect matching and being Hamiltonian, to name a few. In this paper we continue to explore the local resilience notion, but turn our attention to random and pseudo-random \emph{regular} graphs of constant degree. We investigate the local resilience of the typical random $d$-regular graph with respect to edge and vertex connectivity, containing a perfect matching, and being Hamiltonian. In particular we prove that for every positive $\epsilon$ and large enough values of $d$ with high probability the local resilience of the random $d$-regular graph, $\GND$, with respect to being Hamiltonian is at least $(1-\epsilon)d/6$. We also prove that for the Binomial random graph model $\GNP$, for every positive $\epsilon>0$ and large enough values of $K$, if $p>\frac{K\ln n}{n}$ then with high probability the local resilience of $\GNP$ with respect to being Hamiltonian is at least $(1-\epsilon)np/6$. Finally, we apply similar techniques to Positional Games and prove that if $d$ is large enough then with high probability a typical random $d$-regular graph $G$ is such that in the unbiased Maker-Breaker game played on the edges of $G$, Maker has a winning strategy to create a Hamilton cycle.