On Cartesian products of signed graphs

TL;DR

This paper investigates the algebraic properties of Cartesian products of signed graphs, establishes the uniqueness of their prime factorization, and analyzes their chromatic number for various graph classes.

Contribution

It introduces a linear-time algorithm for prime factor decomposition of signed graphs and explores chromatic numbers for specific Cartesian product graph classes.

Findings

Unique prime factor decomposition for signed graphs.

Linear-time algorithm for decomposition.

Chromatic number analysis for various graph classes.

Abstract

In this paper, we study the Cartesian product of signed graphs as defined by Germina, Hameed and Zaslavsky (2011). Here we focus on its algebraic properties and look at the chromatic number of some Cartesian products. One of our main results is the unicity of the prime factor decomposition of signed graphs. This leads us to present an algorithm to compute this decomposition in linear time based on a decomposition algorithm for oriented graphs by Imrich and Peterin (2018). We also study the chromatic number of a signed graph, that is the minimum order of a signed graph to which the input signed graph admits a homomorphism, of graphs with underlying graph of the form P n [] P m , of Cartesian products of signed paths, of Cartesian products of signed complete graphs and of Cartesian products of signed cycles.