On Cartesian Products of Signed Graphs

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 products. One of our main result is the unicity of the prime factor decomposition of signed graphs. This leads us to present an algorithm to compute this decomposition in quasi-linear time. Both these results use their counterparts for ordinary graphs as building blocks. We also study the signed chromatic number of graphs with underlying graph of the form of products of signed paths, of products of signed complete graphs and of products of signed cycles, that is the minimum order of a signed graph to which they admit a homomorphism.