# Tutte's dichromate for signed graphs

## TL;DR

This paper introduces the trivariate Tutte polynomial for signed graphs, unifying various graph invariants and extending to matroid pairs, revealing new relationships and differences with existing polynomials.

## Contribution

It defines a new invariant, the trivariate Tutte polynomial for signed graphs, and explores its properties, extensions, and distinctions from related polynomials.

## Key findings

- The trivariate Tutte polynomial encodes proper colorings and flows of signed graphs.
- It differs from the dichromatic polynomial in capturing nowhere-zero flows.
- The polynomial extends to matroid pairs, linking graph theory and matroid theory.

## Abstract

We introduce the ``trivariate Tutte polynomial" of a signed graph as an invariant of signed graphs up to vertex switching that contains among its evaluations the number of proper colorings and the number of nowhere-zero flows. In this, it parallels the Tutte polynomial of a graph, which contains the chromatic polynomial and flow polynomial as specializations. The number of nowhere-zero tensions (for signed graphs they are not simply related to proper colorings as they are for graphs) is given in terms of evaluations of the trivariate Tutte polynomial at two distinct points. Interestingly, the bivariate dichromatic polynomial of a biased graph, shown by Zaslavsky to share many similar properties with the Tutte polynomial of a graph, does not in general yield the number of nowhere-zero flows of a signed graph. Therefore the ``dichromate" for signed graphs (our trivariate Tutte polynomial) differs from the dichromatic polynomial (the rank-size generating function). The trivariate Tutte polynomial of a signed graph can be extended to an invariant of ordered pairs of matroids on a common ground set -- for a signed graph, the cycle matroid of its underlying graph and its frame matroid form the relevant pair of matroids. This invariant is the canonically defined Tutte polynomial of matroid pairs on a common ground set in the sense of a recent paper of Krajewski, Moffatt and Tanasa, and was first studied by Welsh and Kayibi as a four-variable linking polynomial of a matroid pair on a common ground set.