Discussiones Mathematicae Graph Theory 33(3) (2013) 613-632
doi: 10.7151/dmgt.1693

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Interval edge-colorings of Cartesian products of graphs I

Petros A. Petrosyan

Department of Informatics and Applied Mathematics
Yerevan State University, 0025, Armenia
Institute for Informatics and Automation Problems
National Academy of Sciences, 0014, Armenia

Hrant H. Khachatrian

Department of Informatics and Applied Mathematics
Yerevan State University, 0025, Armenia

Hovhannes G. Tananyan

Department of Applied Mathematics and Informatics
Russian-Armenian State University, 0051, Armenia

Abstract

A proper edge-coloring of a graph G with colors 1, …,t is an interval t-coloring if all colors are used and the colors of edges incident to each vertex of G form an interval of integers. A graph G is interval colorable if it has an interval t-coloring for some positive integer t. Let ℕ be the set of all interval colorable graphs. For a graph G ∈ ℕ, the least and the greatest values of t for which G has an interval t-coloring are denoted by w(G) and W(G), respectively. In this paper we first show that if G is an r-regular graph and G ∈ ℕ, then W(G□Pm) ≥ W(G)+W(Pm)+(m −1)r (m ∈ ℕ) and W(G□C2n) ≥ W(G)+W(C2n)+nr (n ≥ 2). Next, we investigate interval edge-colorings of grids, cylinders and tori. In particular, we prove that if G□H is planar and both factors have at least 3 vertices, then G□H ∈ ℕ and w(G□H) ≤ 6. Finally, we confirm the first author's conjecture on the n-dimensional cube Qn and show that Qn has an interval t-coloring if and only if n ≤ t ≤ n(n+1)/2.

Keywords: edge-coloring, interval coloring, grid, cylinder, torus, n-dimensional cube

2010 Mathematics Subject Classification: 05C15.

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Received 31 January 2012
Revised 5 March 2013
Accepted 5 March 2013