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. Klavar et al. [S. Klavar, A. Lipovec, M. Petkovek, On subgraphs of Cartesian product graphs, Discrete Math. 244 (2002) 223230] 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