Edge-coloring of generalized lexicographic product of graphs

An edge-coloring of a graph $G$ is an assignment of colors to its edges so that no two edges incident to the same vertex receive the same color. The chromatic index of $G$, denoted by ${\chi }^{\prime }(G)$, is the least $k$ for which $G$ has a $k$ edge-coloring. Graphs with ${\chi }^{\prime }(G)=\mathrm{\Delta }(G)$ are said to be Class 1, and graphs with ${\chi }^{\prime }(G)=\mathrm{\Delta }(G)+1$ are said to be Class 2. Let $G$ be a graph with $V(G)=\{t_1,t_2,\dots ,t_n\}$, $n\ge 2$, and $h_n=(H_i{)}_{i\in \{1,2,\dots ,n\}}$ be a sequence of vertex-disjoint graphs with $V(H_i)=\{(t_i,y_1),(t_i,y_2),\dots ,(t_i,y_{m_i})\}$, $m_i\ge 1$. The generalized lexicographic product $G[h_n]$ of $G$ and $h_n=(H_i{)}_{i\in \{1,2,\dots ,n\}}$ is a simple graph with vertex set $\bigcup _{i=1}^{n}V(H_i)$, in which $(t_i,y_p)$ is adjacent to $(t_j,y_q)$ if and only if either $t_i=t_j$ and $(t_i,y_p)(t_i,y_q)\in E(H_i)$ or $t_i{t}_j\in E(G)$. If $G$ is a complete graph with order 2, then $G[h_2]$ denotes a join $H_1+H_2$ of vertex-disjoint graphs $H_1$ and $H_2$. If $H_i\cong H$ for $i=1,2,\dots ,n$, then $G[h_n]=G[H]$, where $G[H]$ denotes the lexicographic product of two graphs $G$ and $H$. In this paper, we provide sufficient conditions for the generalized lexicographic product $G[h_n]$ of $G$ and $h_n=(H_i{)}_{i\in \{1,2,\dots ,n\}}$ to be Class 1, where all graphs in $h_n$ have the same number of vertices.