Interval edge-colorings of cartesian products of graphs I

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 א be the set of all interval colorable graphs. For a graph G ∈ א, 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 ∈ א, then W(G□Pm) ≥ W(G) + W(P m) + (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 ∈ א 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.