The chromatic number of the cartesian product of signed graphs

In 1963, Vizing showed that the chromatic number of the Cartesian product of two graphs is equal to the larger one of the chromatic numbers of these two graphs. This result shows that the chromatic number of the Cartesian product of graphs is determined by its factors’ chromatic numbers. In this paper, we study the chromatic number of the Cartesian product of signed graphs. For signed graphs, we find that the chromatic number of the Cartesian product is also determined by its factors. For any two signed graphs [Formula: see text] and [Formula: see text], we have [Formula: see text]. Moreover, if [Formula: see text] is even, then [Formula: see text]. For the case [Formula: see text] is odd, there are graphs that satisfy [Formula: see text] and [Formula: see text] respectively.