Elsevier

Discrete Mathematics

Volume 312, Issue 1, 6 January 2012, Pages 74-80
Discrete Mathematics

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

https://doi.org/10.1016/j.disc.2011.03.033Get rights and content
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Abstract

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.

Keywords

S-prime
Diagonalized Cartesian product
Path-k-coloring

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