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

For an increasing monotone graph property P the local resilience of a graph G with respect to P 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 P . 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 G(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, Gn,d, with respect to being Hamiltonian, is at least (1− ε)d/6. We also prove that for the binomial random graph model G(n, p), for every positive ε > 0 and large enough values of K , if p > K ln n n then, with high probability, the local resilience of G(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 if d is large enough then, with high probability, a typical