Embedding spanning bounded degree graphs in randomly perturbed graphs

TL;DR

This paper investigates the embedding of bounded degree spanning graphs in randomly perturbed dense graphs, introducing a new absorption-based approach that simplifies proofs and establishes lower thresholds for subgraph appearance.

Contribution

It presents a general absorption method for studying spanning subgraphs in perturbed graphs and derives new lower bounds for embedding bounded degree graphs and Hamilton cycle powers.

Findings

Lower bounds on p for embedding graphs with maximum degree Δ

First example where embedding threshold is significantly lower than in G(n,p)

Threshold for Hamilton cycle powers in perturbed graphs is close to known in G(n,p)

Abstract

We study the model $G_\alpha\cup G(n,p)$ of randomly perturbed dense graphs, where $G_\alpha$ is any $n$-vertex graph with minimum degree at least $\alpha n$ and $G(n,p)$ is the binomial random graph. We introduce a general approach for studying the appearance of spanning subgraphs in this model using absorption. This approach yields simpler proofs of several known results. We also use it to derive the following two new results. For every $\alpha>0$ and $\Delta\ge 5$, and every $n$-vertex graph $F$ with maximum degree at most $\Delta$, we show that if $p=\omega(n^{-2/(\Delta+1)})$ then $G_\alpha \cup G(n,p)$ with high probability contains a copy of $F$. The bound used for $p$ here is lower by a $\log$-factor in comparison to the conjectured threshold for the general appearance of such subgraphs in $G(n,p)$ alone, a typical feature of previous results concerning randomly perturbed dense graphs. We also give the first example of graphs where the appearance threshold in $G_\alpha \cup G(n,p)$ is lower than the appearance threshold in $G(n,p)$ by substantially more than a $\log$-factor. We prove that, for every $k\geq 2$ and $\alpha >0$, there is some $\eta>0$ for which the $k$th power of a Hamilton cycle with high probability appears in $G_\alpha \cup G(n,p)$ when $p=\omega(n^{-1/k-\eta})$. The appearance threshold of the $k$th power of a Hamilton cycle in $G(n,p)$ alone is known to be $n^{-1/k}$, up to a $\log$-term when $k=2$, and exactly for $k>2$.