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 χ i ( G ) . We approach the question how much χ i ( G ) differs from the usual chromatic number χ ( G ) . In particular, whether there is a function f such that χ i ( G ) ⩽ f ( χ ( G ) ) for every graph G . We prove that f 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 χ ( G ) . We also exhibit several classes of graphs for which χ i ( G ) = χ ( G ) and show that this equality for any class of perfect graphs implies Clique-Pair Conjecture for this class of graphs.