Hat chromatic number of graphs

We study the hat chromatic number of a graph defined in following way: there is one player at each vertex G , an adversary places K colors on head player, two players can see other's hats if and only they are adjacent vertices. All simultaneously try to guess color their hat. The cannot communicate but collectively determine strategy before placed. number, denoted by HG ( ) largest such that able fix will ensure for every possible placement least guesses correctly. several classes graphs, others we establish some bounds. investigate connections between coloring number. also introduce variants game: with multiple guesses, restrictions allowed strategies or colorings. show examples how modified games be used obtain interesting results original game.