Interval edge-colorings of composition of graphs

An edge-coloring of a graph G with consecutive integers c1, . . . , ct is called an interval t-coloring if all colors are used, and the colors of edges incident to any vertex of G are distinct and form an interval of integers. A graph G is interval colorable if it has an interval t-coloring for some positive integer t. The set of all interval colorable graphs is denoted by N. In 2004, Giaro and Kubale showed that if G,H ∈ N, then the Cartesian product of these graphs belongs to N. In the same year they formulated a similar problem for the composition of graphs as an open problem. Later, in 2009, the first author showed that if G,H ∈ N and H is a regular graph, then G[H ] ∈ N. In this paper, we prove that if G ∈ N and H has an interval coloring of a special type, then G[H ] ∈ N. Moreover, we show that all regular graphs, complete bipartite graphs and trees have such a special interval coloring. In particular, this implies that if G ∈ N and T is a tree, then G[T ] ∈ N.