On star and acyclic coloring of generalized lexicographic product of graphs

A $starcoloring$ of a graph $G$ is a proper vertex coloring of $G$ such that any path of length 3 in $G$ is not bicolored. The $starchromaticnumber$ ${\chi }_s(G)$ of $G$ is the smallest integer $k$ for which $G$ admits a star coloring with $k$ colors. A $acycliccoloring$ of $G$ is a proper coloring of $G$ such that any cycle in $G$ is not bicolored. The $acyclicchromaticnumber$ of $G$, denoted by $a(G)$, is the minimum number of colors needed to acyclically color $G$. In this paper, we present upper bound for the star and acyclic chromatic numbers of the generalized lexicographic product $G[h_n]$ of graph $G$ and disjoint graph sequence $h_n$, where $G$ exists a $k-$colorful neighbor star coloring or $k-$colorful neighbor acyclic coloring. In addition, the upper bounds are tight.