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 $\mathfrak{N}$ be the set of all interval colorable graphs. For a graph $G\in \mathfrak{N}$, 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 $G\in \mathfrak{N}$, then $W(G\square P_{m})\geq W(G)+W(P_{m})+(m-1)r$ ($m\in \mathbb{N}$) and $W(G\square C_{2n})\geq W(G)+W(C_{2n})+nr$ ($n\geq 2$). Next, we investigate interval edge-colorings of grids, cylinders and tori. In particular, we prove that if $G\square H$ is planar and both factors have at least 3 vertices, then $G\square H\in \mathfrak{N}$ and $w(G\square H)\leq 6$. Finally, we confirm the first author's conjecture on the $n$-dimensional cube $Q_{n}$ and show that $Q_{n}$ has an interval $t$-coloring if and only if $n\leq t\leq \frac{n(n+1)}{2}$.