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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
be any subgraph of the random n-vertex d-regular graph
with minimum degree at least
$$(1/2 + \epsilon )d$$
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.
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