The Matching Process and Independent Process in Random Regular Graphs and Hypergraphs

TL;DR

This paper analyzes two processes on sparse random graphs and hypergraphs with fixed degree sequences, simplifying the differential equations approach to determine the size of matchings and independent sets.

Contribution

It introduces a simplified differential equations framework for analyzing matching and independent processes in sparse random graphs and hypergraphs.

Findings

Reduced the systems of differential equations needed for analysis

Simplified the expression for the final size of matchings and independent sets

Applied Warnke's theorem to streamline the analysis

Abstract

In this note, we analyze two random greedy processes on sparse random graphs and hypergraphs with a given degree sequence. First we analyze the matching process, which builds a set of disjoint edges one edge at a time; then we analyze the independent process, which builds an independent set of vertices one vertex at a time. We use the differential equations method and apply a general theorem of Warnke. Our main contribution is to significantly reduce the associated systems of differential equations and simplify the expression for the final size of the matching or independent set.