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 ε>0, a.a.s. the following holds: let G′ be any
subgraph of the random n-vertex d-regular graph Gn,d with minimum
degree at least (1/2+ε)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.Comment: Final accepted version, to appear in Combinatorics, Probability &
Computin