Chain characteristic polynomials of matroids

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.