Geodesic
Interval edge colorings of some products of graphs
Discussiones Mathematicae. Graph Theory, Tome 31 (2011) no. 2, pp. 357-373

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An edge coloring of a graph G with colors 1,2,...,t is called an interval t-coloring if for each i ∈ 1,2,...,t there is at least one edge of G colored by i, and the colors of edges incident to any vertex of G are distinct and form an interval of integers. A graph G is interval colorable, if there is an integer t ≥ 1 for which G has an interval t-coloring. Let ℜ be the set of all interval colorable graphs. In 2004 Kubale and Giaro showed that if G,H ∈ , then the Cartesian product of these graphs belongs to . Also, they formulated a similar problem for the lexicographic product as an open problem. In this paper we first show that if G ∈ , then G[nK₁] ∈ for any n ∈ ℕ. Furthermore, we show that if G,H ∈ and H is a regular graph, then strong and lexicographic products of graphs G,H belong to . We also prove that tensor and strong tensor products of graphs G,H belong to if G ∈ and H is a regular graph.
Keywords: edge coloring, interval coloring, regular graph, products of graphs 
Petrosyan, Petros. Interval edge colorings of some products of graphs. Discussiones Mathematicae. Graph Theory, Tome 31 (2011) no. 2, pp. 357-373. http://geodesic.mathdoc.fr/item/DMGT_2011_31_2_a11/
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