  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 $\varepsilon>0$, a.a.s. the following holds: let $G'$ be any
subgraph of the random $n$-vertex $d$-regular graph $G_{n,d}$ with minimum
degree at least $(1/2+\varepsilon)d$. Then $G'$ 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.
