Local resilience and hamiltonicity maker-breaker games in random regular graphs

Abstract

For an increasing monotone graph property the local resilience of a graph G with respect to is the minimal r for which there exists a subgraph H G with all degrees at most r, such that the removal of the edges of H from G creates a graph that does not possess. 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 (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-random 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 ε and large enough values of d, with high probability, the local resilience of the random d-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 > 0 and large enough values of K, if p > K lnn/n then, with high probability, the local resilience of (n, p) with respect to being Hamiltonian is at least (1)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. © 2010 Cambridge University Press.