The Property of Having a $k$-Regular Subgraph Has a Sharp Threshold

TL;DR

This paper proves that the property of a random graph containing a $k$-regular subgraph exhibits a sharp threshold for $k ext{ }\ge 3$, and also provides a simple proof for the sharpness of the $k$-core property.

Contribution

It establishes the sharp threshold for the existence of $k$-regular subgraphs in $G(n,p)$ and offers a straightforward proof for the sharpness of the $k$-core property.

Findings

Sharp threshold for $k$-regular subgraphs in $G(n,p)$ for $k\ge3$

Simple proof for the sharpness of the $k$-core property

Methodology applicable to similar threshold phenomena

Abstract

We prove that the property of containing a $k$-regular subgraph in the random graph model $G(n,p)$ has a sharp threshold for $k\ge3$. We also show how to use similar methods to obtain an easy prove for the (known fact of) sharpness of having a non empty $k$-core for $k\ge3$.