On indicated coloring of lexicographic product of graphs

Indicated coloring is a graph coloring game in which two players collectively color the vertices of a graph in the following way. In each round the first player (Ann) selects a vertex, and then the second player (Ben) colors it properly, using a fixed set of colors. The goal of Ann is to achieve a proper coloring of the whole graph, while Ben is trying to prevent the realization of this project. The smallest number of colors necessary for Ann to win the game on a graph G (regardless of Ben's strategy) is called the indicated chromatic number of G, denoted by chi i(G). In this paper, we show that for any graphs G and H, G[H] is k-indicated colorable for all k > col(G)col(H). Also, we show that for any graph G and for some classes of graphs H with chi(H) = chi i(H) = l, G[H] is k-indicated colorable if and only if G[Kl] is k-indicated colorable. As a consequence {of this result, we show that if G E G = Chordal graphs, Cographs, Complement of bipartite graphs, {P5, C-4}-free graphs, Connected {P-6, C-5, ((P-5) over bar), K-1,K-3}-free graphs which } { contain an induced C6, Complete multipartite graphs and H E F = Bipartite graphs, Chordal graphs, Cographs, {P-5, K-3}-free graphs, {P-5, paw}-free graphs, Complement of bipartite graphs, {P-5,K-4, kite, bull}-free graphs, Connected {P-6, C-5, P-5, K-1,K-3}-free graphs which contain an induced C-6, {P-5, C-4}-free graphs, K[C-5](m(1), m(2), ... , m(5)), Connected }{P-5, ((P-2 boolean OR P-3) over bar), ((P-5)over bar), dart}-free graphs which contain an induced C-5 , then G[H] is k-indicated colorable for every k > chi(G[H]). This serves as a partial answer to one of the questions {raised by A. Grzesik in Grzesik (2012). In addition, if G E Bipartite graphs, {P5, K3}-free }graphs, {P5, paw}-free graphs and H E F, then we show that chi(i)(G[H])= chi(G[H]). (C) 2021 Elsevier B.V. All rights reserved.