Large matchings and nearly spanning, nearly regular subgraphs of random subgraphs

TL;DR

This paper demonstrates that in large random subgraphs of regular graphs, nearly spanning matchings and dense subgraphs with concentrated degrees typically exist, extending understanding of random subgraph structures.

Contribution

It establishes conditions under which large matchings and nearly spanning, nearly regular subgraphs appear in random subgraphs of regular graphs, including hypercubes.

Findings

Large matchings cover at least (1-ε) of vertices with high probability

Existence of nearly regular induced subgraphs with degrees concentrated around dp

Results hold for various regular graphs including hypercubes

Abstract

Given a graph $G$ and $p\in [0,1]$, the random subgraph $G_p$ is obtained by retaining each edge of $G$ independently with probability $p$. We show that for every $\epsilon>0$, there exists a constant $C>0$ such that the following holds. Let $d\ge C$ be an integer, let $G$ be a $d$-regular graph and let $p\ge \frac{C}{d}$. Then, with probability tending to one as $|V(G)|$ tends to infinity, there exists a matching in $G_p$ covering at least $(1-\epsilon)|V(G)|$ vertices. We further show that for a wide family of $d$-regular graphs $G$, which includes the $d$-dimensional hypercube, for any $p\ge \frac{\log^5d}{d}$ with probability tending to one as $d$ tends to infinity, $G_p$ contains an induced subgraph on at least $(1-o(1))|V(G)|$ vertices, whose degrees are tightly concentrated around the expected average degree $dp$.