Interval edge-colorings of Cartesian products of graphs I

An 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 are distinct and form an interval of integers. A graph G is interval colorable if G 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 .