Matching polytons

TL;DR

This paper extends the concepts of matchings and vertex covers from finite graphs to graphons, introducing the matching and fractional vertex cover polytons and analyzing their properties and limits.

Contribution

It initiates the study of matching and fractional vertex cover sets in graphons, defining the polytons and exploring their properties and behavior in convergent sequences.

Findings

Characterization of r-partite graphons via absence of certain subgraphs

Symmetric spectrum as a characterization of bipartite graphons

Introduction of matching and fractional vertex cover polytons in graphons

Abstract

Hladky, Hu, and Piguet [Tilings in graphons, preprint] introduced the notions of matching and fractional vertex covers in graphons. These are counterparts to the corresponding notions in finite graphs. Combinatorial optimization studies the structure of the matching polytope and the fractional vertex cover polytope of a graph. Here, in analogy, we initiate the study of the structure of the set of all matchings and of all fractional vertex covers in a graphon. We call these sets the matching polyton and the fractional vertex cover polyton. We also study properties of matching polytons and fractional vertex cover polytons along convergent sequences of graphons. As an auxiliary tool of independent interest, we prove that a graphon is $r$-partite if and only if it contains no graph of chromatic number $r+1$. This in turn gives a characterization of bipartite graphons as those having a symmetric spectrum.