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

中文翻译：

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随机正则图的狄拉克定理

我们在随机正则图的设置中证明了狄拉克定理的“弹性”版本。更准确地说，我们表明，无论何时d与$\epsilon > 0$, a 如下成立。让$G'$是随机数的任何子图n-顶点d- 正则图$G_{n,d}$至少具有最低学位$$(1/2 + \epsilon )d$$. 然后$G'$是哈密顿量。这证明了 Ben-Shimon、Krivelevich 和 Sudakov 的猜想。我们的结果是最好的：首先，条件是dis large 不能省略，其次不能提高最小度数界限。