The Chromatic Number of Joins of Signed Graphs

TL;DR

This paper introduces joins of signed graphs and investigates their chromatic numbers, establishing a new relation similar to classical graph joins but accounting for signed graph properties and a novel deficiency measure.

Contribution

It defines joins of signed graphs and proves an analogue to the classical chromatic number sum theorem, incorporating the concept of deficiency.

Findings

Chromatic number of joins often less than sum of individual chromatic numbers

Introduces the concept of deficiency in signed graph colorings

Establishes bounds and relations for signed graph joins

Abstract

We introduce joins of signed graphs and explore the chromatic number of the all-positive and all-negative joins. We prove an analogue to the theorem that the chromatic number of the join of two graphs equals the sum of their chromatic numbers. Given two signed graphs, the chromatic number of the all-positive and all-negative join is usually less than the sum of their chromatic numbers, by an amount that depends on the new concept of deficiency of a signed-graph coloration.