  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.
