Indicated coloring of the Mycielskian of some families of graphs

Abstract

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 [Formula: see text]. In this paper, we examine whether the Mycielskian of G, [Formula: see text], is k-indicated colorable for all [Formula: see text], whenever G is l-indicated colorable for all [Formula: see text]. In this direction, we prove that the Mycielskian of the bipartite graphs, complete multipartite graphs, [Formula: see text]-free graphs, [Formula: see text]-free graphs, [Formula: see text]-free graphs and [Formula: see text]-free graphs are k-indicated colorable for all k greater than or equal to the indicated chromatic number of the corresponding graphs.