Geodesic
Adjacent vertex distinguishing edge colorings of the direct product of a regular graph by a path or a cycle
Discussiones Mathematicae. Graph Theory, Tome 31 (2011) no. 3, pp. 547-557

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In this paper we investigate the minimum number of colors required for a proper edge coloring of a finite, undirected, regular graph G in which no two adjacent vertices are incident to edges colored with the same set of colors. In particular, we study this parameter in relation to the direct product of G by a path or a cycle.
Keywords: chromatic index, adjacent vertex distinguishing edge coloring, direct product, matching 
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     title = {Adjacent vertex distinguishing edge colorings of the direct product of a regular graph by a path or a cycle},
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Frigerio, Laura; Lastaria, Federico; Salvi, Norma. Adjacent vertex distinguishing edge colorings of the direct product of a regular graph by a path or a cycle. Discussiones Mathematicae. Graph Theory, Tome 31 (2011) no. 3, pp. 547-557. http://geodesic.mathdoc.fr/item/DMGT_2011_31_3_a9/