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

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 $k\ge 3$. 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 $k\ge 3$.