# DMGT

ISSN 1234-3099 (print version)

ISSN 2083-5892 (electronic version)

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SCImago Journal Rank (SJR) 2018: 0.763

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# Discussiones Mathematicae Graph Theory

## INTERVAL EDGE COLORINGS OF SOME PRODUCTS OF GRAPHS

 Petros A. Petrosyan Institute for Informatics and Automation Problems National Academy of Sciences, 0014, Armenia Department of Informatics and Applied Mathematics Yerevan State University, 0025, Armenia e-mail: pet_petros@{ipia.sci.am, ysu.am, yahoo.com}

## Abstract

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 ∈ ℜ, then the Cartesian product of these graphs belongs to ℜ. Also, they formulated a similar problem for the lexicographic product as an open problem. In this paper we first show that if G ∈ ℜ, then G[nK1] ∈ ℜ for any n ∈ ℕ. Furthermore, we show that if G,H ∈ ℜ and H is a regular graph, then strong and lexicographic products of graphs G,H belong to ℜ. We also prove that tensor and strong tensor products of graphs G,H belong to ℜ if G ∈ ℜ and H is a regular graph.

Keywords: edge coloring, interval coloring, regular graph, products of graphs.

2010 Mathematics Subject Classification: Primary: 05C15,
05C76; Secondary: 05C70.

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