Diagonalized Cartesian products of -prime graphs are -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. [S. Klavžar, A. Lipovec, M. Petkovšek, On subgraphs of Cartesian product graphs, Discrete Math. 244 (2002) 223–230] 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.