Embedding large graphs into a random graph

TL;DR

This paper proves that large bounded degree graphs can be embedded into random graphs with high probability, establishing near-optimal conditions on the edge probability for such embeddings.

Contribution

It provides a near-optimal threshold for embedding almost-spanning bounded degree graphs into random graphs, advancing understanding of graph embedding thresholds.

Findings

Embedding threshold matches conjectured spanning case

High probability embedding under specified p

Optimal up to polylog factors

Abstract

In this paper we consider the problem of embedding almost-spanning, bounded degree graphs in a random graph. In particular, let $\Delta\geq 5$, $\varepsilon > 0$ and let $H$ be a graph on $(1-\varepsilon)n$ vertices and with maximum degree $\Delta$. We show that a random graph $G_{n,p}$ with high probability contains a copy of $H$, provided that $p\gg (n^{-1}\log^{1/\Delta}n)^{2/(\Delta+1)}$. Our assumption on $p$ is optimal up to the $polylog$ factor. We note that this $polylog$ term matches the conjectured threshold for the spanning case.