Diagonalized Cartesian products of s-prime graphs are s-prime

A graph is said to be S-prime if, whenever it is a subgraph of a nontrivial Cartesian product graph, it is a subgraph of one of the factors. A diagonalized Cartesian product is obtained from a Cartesian product graph by connecting two vertices of maximal distance by an additional edge. We show there that a diagonalized product of S-prime graphs is again S-prime. Klavžar et al. [Discr. Math. 244: 223-230 (2002)] proved that a graph is S-prime if and only if it admits a nontrivial path-k-coloring. We derive here a characterization of all path-k-colorings of Cartesian products of S-prime graphs.