Chain characteristic polynomials of matroids

TL;DR

This paper introduces a new family of polynomials for matroids called chain characteristic polynomials, which generalize the classic characteristic polynomial and relate to graph colorings and flows.

Contribution

It defines chain characteristic polynomials for matroids, explores their properties, and connects them to generalized graph colorings and flows.

Findings

Coefficients of chain characteristic polynomials alternate.

Recursion formulas for these polynomials are established.

They enumerate coupled multicolorings and multicommodity flows on graphs.

Abstract

In this note we introduce a family of polynomials on a matroid derived from chain Tutte polynomials which generalize the classic and ubiquitous characteristic polynomial. We show that the coefficients of these polynomials alternate and present a recursion for any matroid. The classic characteristic polynomials were motivated by the chromatic polynomial on graphs. We define a generalized proper vertex coloring on a graph, which we call coupled multicoloring. We also define a generalized nowhere zero flow, which we call coupled multicommodity flow. Then we show that these chain characteristic polynomials enumerate the number of coupled multicolorings and coupled multicommodity flows on a graph. We conclude by listing multiple problems on enumerative properties of chain characteristic polynomials.