Triangle resilience of the square of a Hamilton cycle in random graphs

Abstract

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 $p \geq C\log^3 n/\sqrt{n}$ 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.