Indicated coloring 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 [Formula: see text] (regardless of Ben’s strategy) is called the indicated chromatic number of [Formula: see text], denoted by [Formula: see text]. In this paper, we observe that the Nordhaus–Gaddum inequalities for the chromatic number are also satisfied by the indicated chromatic number. Further, we prove that outerplanar graphs, 3-colorable maximal planar graphs with [Formula: see text], uniquely 4-colorable planar graphs and [Formula: see text] (obtained from [Formula: see text] by joining diagonally opposite vertices) are [Formula: see text]-indicated colorable for all [Formula: see text] greater than or equal to its chromatic number. Also, we show that Ben wins the game with [Formula: see text] colors when Ann uses any connected strategy on [Formula: see text] for any [Formula: see text]. This turns out to be a generalization of the result due to Grzesik in [Indicated coloring of graphs, Discrete Math. 312(23) (2012) 3467–3472]. Finally, we prove that Mycielskian of [Formula: see text] is [Formula: see text]-indicated colorable when [Formula: see text] is [Formula: see text]-indicated colorable for some [Formula: see text]. These partially answer one of the questions which was raised by Grzesik in [Indicated coloring of graphs, Discrete Math. 312(23) (2012) 3467–3472].