An interval t-coloring of a graph G is a proper edge-coloring with colors \(1,\dots ,t\) such that the colors on the edges incident to every vertex of G are colored by consecutive colors. A graph G is called interval colorable if it has an interval t-coloring for some positive integer t. Let \(\mathfrak {N}\) be the set of all interval colorable graphs. For a graph \(G\in \mathfrak {N}\) , we denote by w(G) and W(G) the minimum and maximum number of colors in an interval coloring of a graph G, respectively. In this paper we present some new sharp bounds on \(W(G\square H)\) for graphs G and H satisfying various conditions. In particular, we show that if \(G,H\in \mathfrak {N}\) and H is an r-regular ( \(r\in \mathbb {N}\) ) graph, then \(W(G\square H)\ge W(G)+W(H)+r\) . We also derive a new upper bound on W(G) for interval colorable connected graphs with additional distance conditions. Based on these bounds, we improve known lower and upper bounds on \(W(C_{2n_{1}}\square C_{2n_{2}}\square \cdots \square C_{2n_{k}})\) for k-dimensional tori \(C_{2n_{1}}\square C_{2n_{2}}\square \cdots \square C_{2n_{k}}\) and on \(W(K_{2n_{1}}\square K_{2n_{2}}\square \cdots \square K_{2n_{k}})\) for Hamming graphs \(K_{2n_{1}}\square K_{2n_{2}}\square \cdots \square K_{2n_{k}}\) , and these new bounds coincide with each other for hypercubes. Finally, we give several results on interval colorings of Fibonacci cubes \(\varGamma _{n}.\)
