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.