Interval edge colorings of the generalized lexicographic product of some graphs

An edge-coloring of a graph $G$ with colors $1,\dots ,t$ 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 it has an interval $t$-coloring for some positive integer $t$. For an interval colorable graph $G$, the least and the greatest values of $t$ for which $G$ has an interval $t$-coloring are denoted by $w(G)$ and $W(G)$. Let $G$ be a graph with vertex set $V(G)=\{u_1,\dots ,u_m\}$, $m\ge 2$, and let $h_m=(H_i{)}_{i\in \{1,\dots ,m\}}$ be a sequence of vertex-disjoint with $V(H_i)=\{x_1^{(i)},\dots ,x_{n_i}^{(i)}\}$, $n_i\ge 1$. The generalized lexicographic products $G[h_m]$ of $G$ and $h_m$ is a simple graph with vertex set ${\cup }_{i=1}^{m}V(H_i)$, in which $x_p^{(i)}$ is adjacent to $x_q^{(j)}$ if and only if either $u_i=u_j$ and $x_p^{(i)}{x}_q^{(i)}\in E(H_i)$ or $u_i{u}_j\in E(G)$. In this paper, we obtain several sufficient conditions for the generalized lexicographic product $G[h_m]$ to have interval colorings. We also present some sharp bounds on $w(G[h_m])$ and $W(G[h_m])$ of $G[h_m]$.