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 n vertices with minimum degree at least n / 2 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 p , and prove that there exists a constant C C such that if p ≥ C log ⁡ n / n p ≥ C log n / n , then a.a.s. the resulting random subgraph is still Hamiltonian. Second, we prove that if a ( 1 : b ) (1:b) Maker-Breaker game is played on a Dirac graph, then Maker can construct a Hamiltonian subgraph as long as the bias b b is at most c n / log ⁡ n cn /log n for some absolute constant c > 0 c > 0 . Both of these results are tight up to a constant factor, and are proved under one general framework.