Robust Hamiltonicity of Dirac graphs

A graph is Hamiltonian if it contains a cycle which passes through every vertex of the graph exactly once. A classical theorem of Dirac from 1952 asserts that every graph on $n$ vertices with minimum degree at least $n/2$ is Hamiltonian. We refer to such graphs as Dirac graphs. In this paper we extend Dirac's theorem in two directions and show that Dirac graphs are robustly Hamiltonian in a very strong sense. First, we consider a random subgraph of a Dirac graph obtained by taking each edge independently with probability $p$, and prove that there exists a constant $C$ such that if $p \ge C \log n / n$, then a.a.s. the resulting random subgraph is still Hamiltonian. Second, we prove that if a $(1:b)$ Maker-Breaker game is played on a Dirac graph, then Maker can construct a Hamiltonian subgraph as long as the bias $b$ is at most $cn /\log n$ for some absolute constant $c > 0$. Both of these results are tight up to a constant factor, and are proved under one general framework.