Hamiltonicity in random graphs is born resilient

Abstract

Let {GM}M≥0 be the random graph process, where G0 is the empty graph on n vertices and subsequent graphs in the sequence are obtained by adding a new edge uniformly at random. For each ε>0, we show that, almost surely, any graph GM with minimum degree at least 2 is not only Hamiltonian (as shown independently by Bollobás, and Ajtai, Komlós and Szemerédi), but remains Hamiltonian despite the removal of any set of edges, as long as at most (1/2−ε) of the edges incident to each vertex are removed. We say that such a graph is (1/2−ε)-resiliently Hamiltonian. Furthermore, for each ε>0, we show that, almost surely, each graph GM is not (1/2+ε)-resiliently Hamiltonian. These results strengthen those by Lee and Sudakov on the likely resilience of Hamiltonicity in the binomial random graph.For each k, we denote by G(k) the (possibly empty) maximal subgraph with minimum degree at least k of a graph G. That is, the k-core of G. Krivelevich, Lubetzky and Sudakov have shown that, for each k≥15, in almost every random graph process {GM}M≥0, every non-empty k-core is Hamiltonian. We show that, for each ε>0 and k≥k0(ε), in almost every random graph process {GM}M≥0, every non-empty k-core is (1/2−ε)-resiliently Hamiltonian, but not (1/2+ε)-resiliently Hamiltonian.Results on resilience of Hamiltonicity in the random graph process have been independently shown by Nenadov, Steger and Trujić.