Indicated coloring of graphs

We study 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 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 denoted by chi(i)(G). We approach the question how much chi(i)(G) differs from the usual chromatic number chi(G). In particular, whether there is a function f such that chi(i)(G) <= f (chi(G)) for every graph G. We prove that! cannot be linear with leading coefficient less than 4/3. On the other hand, we show that the indicated chromatic number of random graphs is bounded roughly by 4 chi (G). We also exhibit several classes of graphs for which chi(i)(G) = chi(G) and show that this equality for any class of perfect graphs implies Clique-Pair Conjecture for this class of graphs. (C) 2012 Elsevier B.V. All rights reserved.