On the threshold for k-regular subgraphs of random graphs

TL;DR

This paper proves that for large enough k, the (k+2)-core of a random graph almost surely contains a spanning k-regular subgraph, pinpointing the threshold for the emergence of such subgraphs.

Contribution

It establishes that the threshold for the appearance of a k-regular subgraph coincides with the threshold for the (k+2)-core in random graphs for large k.

Findings

The (k+2)-core of a random graph contains a spanning k-regular subgraph asymptotically almost surely.

The threshold for the emergence of a k-regular subgraph is at most the threshold for the (k+2)-core.

The threshold window for the appearance of a k-regular subgraph is roughly 2/n for large n and k.

Abstract

The $k$-core of a graph is the largest subgraph of minimum degree at least $k$. We show that for $k$ sufficiently large, the $(k + 2)$-core of a random graph $\G(n,p)$ asymptotically almost surely has a spanning $k$-regular subgraph. Thus the threshold for the appearance of a $k$-regular subgraph of a random graph is at most the threshold for the $(k+2)$-core. In particular, this pins down the point of appearance of a $k$-regular subgraph in $\G(n,p)$ to a window for $p$ of width roughly $2/n$ for large $n$ and moderately large $k$.