Since first introduced by Sudakov and Vu in 2008, the study of resilience problems in random graphs received a lot of attention in probabilistic combinatorics. Of particular interest are resilience problems of spanning structures. It is known that for spanning structures which contain many triangles, local resilience cannot prevent an adversary from destroying all copies of the structure by removing a negligible amount of edges incident to every vertex. In this paper we generalise the notion of local resilience to H-resilience and demonstrate its usefulness on the containment problem of the square of a Hamilton cycle. In particular, we show that there exists a constant $C>0$ such that if $p⩾C{\log }^{3}n/\sqrt{n}$ then w.h.p. in every subgraph G of a random graph $G_{n,p}$ there exists the square of a Hamilton cycle, provided that every vertex of G remains on at least a $(4/9+o(1))$-fraction of its triangles from $G_{n,p}$. The constant 4/9 is optimal and the value of p slightly improves on the best-known appearance threshold of such a structure and is optimal up to the logarithmic factor.

自 2008 年由 Sudakov 和 Vu 首次提出以来，随机图中弹性问题的研究在概率组合学中受到了很多关注。特别令人感兴趣的是跨越结构的弹性问题。众所周知，对于包含许多三角形的跨越结构，局部弹性无法通过移除与每个顶点相关的可忽略数量的边来防止对手破坏结构的所有副本。在本文中，我们将局部弹性的概念推广到H弹性，并证明了它在哈密顿循环平方的包含问题上的有用性。特别地，我们证明存在一个常数$C>0$ 这样如果 $磷⩾C{\mathrm{日志}}^{3}n/\sqrt{n}$然后 whp 在随机图的每个子图G中$G_{n,磷}$存在哈密顿圈的平方，前提是G 的每个顶点都至少保持在一个$(4/9+○(1))$- 其三角形的分数来自 $G_{n,磷}$. 常数 4/9 是最佳的，p 的值在这种结构的最著名的出现阈值上略有改进，并且在达到对数因子时是最佳的。