Indicated coloring of the Mycielskian of some families 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 χ i (G). In this paper, we examine whether the Mycielskian of G, μ (G) , is k-indicated colorable for all k ≥ χ i (μ (G)) , whenever G is l-indicated colorable for all l ≥ χ i (G). In this direction, we prove that the Mycielskian of the bipartite graphs, complete multipartite graphs, { P 5 , K 3 } -free graphs, { P 5 , K 4 , K i t e , B u l l } -free graphs, { 2 K 2 , C 4 } -free graphs and { P 6 , C 5 , P 5 ¯ , K 1 , 3 } -free graphs are k-indicated colorable for all k greater than or equal to the indicated chromatic number of the corresponding graphs. [ABSTRACT FROM AUTHOR]