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Decomposable polymatroids and connections with graph coloring

Joseph E. Bonin, Carolyn Chun

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.

Decomposable polymatroids and connections with graph coloring

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.

Paper Structure

This paper contains 5 sections, 21 theorems, 42 equations, 2 figures.

Table of Contents

  1. Introduction
  2. A construction of polymatroids from hypergraphs
  3. Chromatic numbers and chromatic polynomials
  4. Duals and chromatic numbers
  5. $k$-quotient polymatroids

Key Result

Lemma 2.2

Let $D=\{M_i\,:\,i\in[k]\}$ be a decomposition of a polymatroid $\rho$ on $E$.

Figures (2)

  • Figure 1: A $2$-polymatroid counterpart of the Vámos matroid.
  • Figure 2: (a) The rank increases in the polymatroid $\rho_A$ that is defined in Theorem \ref{['thm:quotkpolyExMin']}. (b) The polymatroid arising from $\{0,1,2\}$.

Theorems & Definitions (43)

  • Definition 1.1
  • Example 1
  • Example 2
  • Example 3
  • Definition 2.1
  • Lemma 2.2
  • proof
  • Example 4
  • Theorem 2.3
  • Lemma 2.4
  • ...and 33 more