Chromatic polynomials of signed graphs and dominating-vertex deletion formulae

TL;DR

This paper explores the properties of chromatic polynomials in signed graphs, introduces new bivariate polynomials, and provides recursive deletion formulas, advancing understanding of graph isomorphism and polynomial invariants.

Contribution

It presents non-isomorphic signed graphs with identical chromatic polynomials, derives closed-form formulas for joins of signed complete graphs, and introduces bivariate chromatic polynomials with recursive deletion formulas.

Findings

Existence of non-switching-isomorphic graphs sharing chromatic polynomials

Closed-form expressions for joins of signed complete graphs

Isomorphism characterized by bivariate chromatic polynomials in certain families

Abstract

We exhibit non-switching-isomorphic signed graphs that share a common underlying graph and common chromatic polynomials, thereby answering a question posed by Zaslavsky. For various joins of all-positive or all-negative signed complete graphs, we derive a closed-form expression for their chromatic polynomials. As a generalisation of the chromatic polynomials for a signed graph, we introduce a new pair of bivariate chromatic polynomials. We establish recursive dominating-vertex deletion formulae for these bivariate chromatic polynomials. Finally, we show that for certain families of signed threshold graphs, isomorphism is equivalent to the equality of bivariate chromatic polynomials.