Dirac's theorem for random regular graphs

We prove a `resilience’ version of Dirac’s theorem in the setting of random regular graphs. More precisely, we show that, whenever $d$ is sufficiently large compared to $\epsilon >0$, a.a.s. the following holds: let $G^{\mathrm{\prime }}$ be any subgraph of the random $n$-vertex $d$-regular graph $G_{n,d}$ with minimum degree at least $(1\mathrm{/}2+\epsilon )d$. Then $G^{\mathrm{\prime }}$ is Hamiltonian. This proves a conjecture of Ben-Shimon, Krivelevich and Sudakov. Our result is best possible: firstly, the condition that $d$ is large cannot be omitted, and secondly, the minimum degree bound cannot be improved.