Local Resilience and Hamiltonicity Maker–Breaker Games in Random Regular Graphs

Abstract

For an increasing monotone graph propertythelocal resilienceof a graphGwith respect tois the minimalrfor which there exists a subgraphH⊆Gwith all degrees at mostr, such that the removal of the edges ofHfromGcreates a graph that does not possess. This notion, which was implicitly studied for somead hocproperties, 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(n, p) 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-randomregulargraphs of constant degree. We investigate the local resilience of the typical randomd-regular graph with respect to edge and vertex connectivity, containing a perfect matching, and being Hamiltonian. In particular, we prove that for every positive ϵ and large enough values ofd, with high probability, the local resilience of the randomd-regular graph,n, d, with respect to being Hamiltonian, is at least (1−ϵ)d/6. We also prove that for the binomial random graph model(n, p), for every positive ϵ > 0 and large enough values ofK, ifp>$\frac{K\ln n}{n}$then, with high probability, the local resilience of(n, p) with respect to being Hamiltonian is at least (1−ϵ)np/6. Finally, we apply similar techniques to positional games, and prove that ifdis large enough then, with high probability, a typical randomd-regular graphGis such that, in the unbiased Maker–Breaker game played on the edges ofG, Maker has a winning strategy to create a Hamilton cycle.