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 G (regardless of Ben's strategy) is called the indicated chromatic number of G, denoted by chi(i)(G). 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 col(G)not equal 5, uniquely 4-colorable planar graphs and M-n (obtained from Cn by joining diagonally opposite vertices) are k-indicated colorable for all k greater than or equal to its chromatic number. Also, we show that Ben wins the game with 4 colors when Ann uses any connected strategy on M4t+1 for any t >= 2. 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 G is (k + 1)-indicated colorable when G is k-indicated colorable for some k >= chi i(G). These partially answer one of the questions which was raised by Grzesik in [Indicated coloring of graphs, Discrete Math. 312(23) (2012) 3467-3472].