The chromatic number of a signed graph

TL;DR

This paper introduces a new definition of chromatic number for signed graphs, extending classical graph coloring concepts, and explores its properties, bounds, and specific cases like planar graphs.

Contribution

It proposes a natural extension of the chromatic number for signed graphs and establishes foundational properties and bounds, including an extension of Brooks' theorem.

Findings

Defined a chromatic number for signed graphs

Established bounds relating to underlying unsigned graphs

Extended Brooks' theorem to signed graphs

Abstract

In 1982, Zaslavsky introduced the concept of a proper vertex colouring of a signed graph $G$ as a mapping $\phi\colon V(G)\to \mathbb{Z}$ such that for any two adjacent vertices $u$ and $v$ the colour $\phi(u)$ is different from the colour $\sigma(uv)\phi(v)$, where is $\sigma(uv)$ is the sign of the edge $uv$. The substantial part of Zaslavsky's research concentrated on polynomial invariants related to signed graph colourings rather than on the behaviour of colourings of individual signed graphs. We continue the study of signed graph colourings by proposing the definition of a chromatic number for signed graphs which provides a natural extension of the chromatic number of an unsigned graph. We establish the basic properties of this invariant, provide bounds in terms of the chromatic number of the underlying unsigned graph, investigate the chromatic number of signed planar graphs, and prove an extension of the celebrated Brooks theorem to signed graphs.