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Interval edge colorings of some products of graphs

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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 ∈ 𝔑, then the Cartesian product of these graphs belongs to 𝔑. Also, they formulated a similar problem for the lexicographic product as an open problem. In this paper we first show that if G ∈ 𝔑, then G[nK₁] ∈ 𝔑 for any n ∈ ℕ. Furthermore, we show that if G,H ∈ 𝔑 and H is a regular graph, then strong and lexicographic products of graphs G,H belong to 𝔑. We also prove that tensor and strong tensor products of graphs G,H belong to 𝔑 if G ∈ 𝔑 and H is a regular graph.
Opis fizyczny
  • Institute for Informatics and Automation Problems, National Academy of Sciences, 0014, Armenia
  • Department of Informatics and Applied Mathematics, Yerevan State University, 0025, Armenia
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