On star and acyclic coloring of generalized lexicographic product of graphs

Abstract

<abstract><p>A $ star \; coloring $ of a graph $ G $ is a proper vertex coloring of $ G $ such that any path of length 3 in $ G $ is not bicolored. The $ star \; chromatic \; number $ $ \chi_s(G) $ of $ G $ is the smallest integer $ k $ for which $ G $ admits a star coloring with $ k $ colors. A $ acyclic \; coloring $ of $ G $ is a proper coloring of $ G $ such that any cycle in $ G $ is not bicolored. The $ acyclic \; chromatic \; number $ 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.</p></abstract>