Geodesic
Proper Connection Of Direct Products
Discussiones Mathematicae. Graph Theory, Tome 37 (2017) no. 4, pp. 1005-1013

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The proper connection number of a graph is the least integer k for which the graph has an edge coloring with k colors, with the property that any two vertices are joined by a properly colored path. We prove that given two connected non-bipartite graphs, one of which is (vertex) 2-connected, the proper connection number of their direct product is 2.
Keywords: direct product of graphs, proper connection of graphs 
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Hammack, Richard H.; Taylor, Dewey T. Proper Connection Of Direct Products. Discussiones Mathematicae. Graph Theory, Tome 37 (2017) no. 4, pp. 1005-1013. http://geodesic.mathdoc.fr/item/DMGT_2017_37_4_a10/