On Indicated Coloring of Some Classes of Graphs

Abstract

Indicated coloring is a type of game coloring 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 obtain structural characterization of $\{P_5,K_4,Kite,Bull\}$-free graphs and connected $\{P_6,C_5, K_{1,3}\}$-free graphs that contain an induced $C_6$. Also, we prove that $\{P_5,K_4,Kite,Bull\}$-free graphs and connected $\{P_6,C_5,\overline{P_5}, K_{1,3}\}$-free graphs which contain an induced $C_6$ are k-indicated colorable for all $k\ge \chi (G)$. In addition, we show that, if $m\ge 1$ and G is a bipartite graph, then $\chi _i(\mathbb {K}[G](m,m,\ldots ,m))=\chi (\mathbb {K}[G](m,m,\ldots ,m))$. Further, we show that $\mathbb {K}[C_5]$ is k-indicated colorable for all $k\ge \chi (G)$ and as a consequence, we exhibit that $\{P_2\cup P_3, C_4\}$-free graphs, $\{P_5,C_4\}$-free graphs are k-indicated colorable for all $k\ge \chi (G)$. This partially answers one of the questions which was raised by Grzesik (Discret Math 312:3467–3472, 2012).