Discussiones Mathematicae Graph Theory 33(3) (2013)
613-632
DOI: https://doi.org/10.7151/dmgt.1693
Interval edge-colorings of Cartesian products of graphs I
Petros A. Petrosyan
Department of Informatics and Applied Mathematics | Hrant H. Khachatrian
Department of Informatics and Applied Mathematics | Hovhannes G. Tananyan
Department of Applied Mathematics and Informatics |
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
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