The perturbation threshold of degenerate graphs

TL;DR

This paper establishes thresholds for embedding degenerate graphs into dense graphs combined with sparse random graphs, revealing conditions under which such embeddings almost surely occur.

Contribution

It introduces new probabilistic thresholds for embedding degenerate and regular graphs into dense graphs plus sparse random graphs, extending previous results.

Findings

Embedding thresholds depend on degeneracy and maximum degree.

Almost all d-regular graphs satisfy the expansion property used.

Thresholds are improved for d-regular graphs with expansion properties.

Abstract

We show that for any $d\ge 2$ and $\Delta>0$ there exists $\eta>0$ such that the following holds: Let $G$ be an $n$-vertex graph with at least $\Omega(n^2)$ edges and let $H$ be an $n$-vertex $d$-degenerate graph with maximum degree at most $\Delta$. Then with high probability, $G \cup G(n, n^{-1/d - \eta})$ contains a copy of $H$. We also prove that the same conclusion extends to $d$-regular graphs with $d\ge 4$ satisfying a certain edge expansion property, with the threshold improved to $n^{-2/d - \eta}$. Such a property is satisfied by almost all $d$-regular graphs and for even $d$, by the $(d/2)$-th power of a Hamilton cycle.