Decomposable polymatroids and connections with graph coloring

TL;DR

This paper explores the relationship between polymatroids and graph coloring, introducing new concepts like chromatic numbers and polynomials for polymatroids, and establishing connections with matroid sums and hypergraph properties.

Contribution

It introduces a framework linking polymatroids to graph coloring via matroid sums, and defines chromatic polynomials for polymatroids, expanding the theoretical understanding of these structures.

Findings

Polymatroids can be constructed from hypergraphs with chromatic number equal to the minimum matroid sum.

Chromatic polynomial of any 2-polymatroid is a rational multiple of some graph's chromatic polynomial.

Identifies excluded minors for a class of polymatroids related to matroid chains.

Abstract

We introduce ideas that complement the many known connections between polymatroids and graph coloring. Given a hypergraph that satisfies certain conditions, we construct polymatroids, given as rank functions, that can be written as sums of rank functions of matroids, and for which the minimum number of matroids required in such sums is the chromatic number of the line graph of the hypergraph. This result motivates introducing chromatic numbers and chromatic polynomials for polymatroids. We show that the chromatic polynomial of any 2-polymatroid is a rational multiple of the chromatic polynomial of some graph. We also find the excluded minors for the minor-closed class of polymatroids that can be written as sums of rank functions of matroids that form a chain of quotients.