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

TL;DR

This paper introduces a new resilience concept called $H$-resilience and applies it to prove the existence of the square of a Hamilton cycle in random graphs under certain local triangle-preservation conditions, improving known thresholds.

Contribution

It generalizes local resilience to $H$-resilience and demonstrates its effectiveness in establishing the containment of the square of a Hamilton cycle in random graphs.

Findings

Existence of the square of a Hamilton cycle under $H$-resilience conditions.

Improved threshold for the appearance of the square of a Hamilton cycle in random graphs.

Optimality of the triangle-preservation fraction $4/9$.

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.