k-regular subgraphs near the k-core threshold of a random graph

TL;DR

This paper establishes the precise threshold for the emergence of k-regular subgraphs in random graphs near the k-core threshold, showing they appear with high probability once the average degree exceeds a certain exponential function of k.

Contribution

It precisely determines the threshold window for the appearance of k-regular subgraphs in Erdős–Rényi random graphs near the k-core threshold.

Findings

k-regular subgraphs appear whp above the threshold c ≥ e^{- heta(k)}

Threshold for k-regular subgraph is within a window of size e^{- heta(k)}

Pinpoints the exact emergence window for k-regular subgraphs near the k-core threshold

Abstract

We prove that $G_{n,p=c/n}$ whp has a $k$-regular subgraph if $c$ is at least $e^{-\Theta(k)}$ above the threshold for the appearance of a subgraph with minimum degree at least $k$; i.e. an non-empty $k$-core. In particular, this pins down the threshold for the appearance of a $k$-regular subgraph to a window of size $e^{-\Theta(k)}$.