On Indicated Coloring of Graphs

Abstract

Indicated coloring is a graph coloring game in which there are two players collectively coloring 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 $G$, 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 $G$, and is denoted by $\chi _i(G)$. In this paper, we have shown that cographs, chordal graphs, complement of bipartite graphs, $\{P_5,K_3\}$-free graphs and $\{P_5, \mathrm{paw}\}$-free graphs are $k$-indicated colorable for all $k\ge \chi _i(G)$. This provides a partial answer to a question raised in Grzesik (Discret Math 312:3467–3472, 2012). Also we have discussed the Brooks’ type result for indicated coloring.