Chromatic polynomials of 2-edge coloured graphs

TL;DR

This paper extends the concept of chromatic polynomials to 2-edge-coloured graphs, providing formulas for coefficients, classifying special cases, and analyzing root behaviors distinct from traditional graph chromatic polynomials.

Contribution

It introduces a new chromatic polynomial for 2-edge-coloured graphs, derives closed-form coefficients, and explores root behaviors unique to this generalization.

Findings

Closed-form expressions for the first three coefficients.

Classification of graphs with matching chromatic polynomials.

Distinct root behaviors compared to standard chromatic polynomials.

Abstract

Using the definition of colouring of $2$-edge-coloured graphs derived from $2$-edge-coloured graph homomorphism, we extend the definition of chromatic polynomial to $2$-edge-coloured graphs. We find closed forms for the first three coefficients of this polynomial that generalize the known results for the chromatic polynomial of a graph. We classify those $2$-edge-coloured graphs that have a chromatic polynomial equal to the chromatic polynomial of the underlying graph, when every vertex is incident to edges of both colours. Finally, we examine the behaviour of the roots of this polynomial, highlighting behaviours not seen in chromatic polynomials of graphs.