On a k-matching algorithm and finding k-factors in random graphs with minimum degree k+1 in linear time

TL;DR

This paper proves that in certain random graphs with minimum degree k+1, a near perfect k-matching exists with high probability, and provides a linear-time algorithm to find such matchings, resolving a longstanding conjecture.

Contribution

It establishes the existence of near perfect k-matchings in random graphs with minimum degree k+1 and introduces a linear-time algorithm to find them, resolving a key conjecture.

Findings

Near perfect k-matching exists with high probability in specified random graphs.

A linear-time algorithm can find the k-matching in these graphs.

The results improve previous bounds and resolve a conjecture in the field.

Abstract

We prove that for $k+1\geq 3$ and $c>(k+1)/2$ w.h.p. the random graph on $n$ vertices, $cn$ edges and minimum degree $k+1$ contains a (near) perfect $k$-matching. As an immediate consequence we get that w.h.p. the $(k+1)$-core of $G_{n,p}$, if non empty, spans a (near) spanning $k$-regular subgraph. This improves upon a result of Chan and Molloy and completely resolves a conjecture of Bollob\'as, Kim and Verstra\"{e}te. In addition, we show that w.h.p. such a subgraph can be found in linear time. A substantial element of the proof is the analysis of a randomized algorithm for finding $k$-matchings in random graphs with minimum degree $k+1$.