Game-perfect digraphs

摘要

In the A-coloring game, two players, Alice and Bob, color uncolored vertices of a given uncolored digraph D with colors from a given color set C, so that, at any time a vertex is colored, its color has to be different from the colors of its previously colored in-neighbors. Alice begins. The players move alternately, where a move of Bob consists in coloring a vertex, and a move of Alice in coloring a vertex or missing the turn. The game ends when Bob is unable to move. Alice wins if every vertex is colored at the end, otherwise Bob wins. This game is a variant of a graph coloring game proposed by Bodlaender (Int J Found Comput Sci 2:133–147, 1991). The A-game chromatic number of D is the smallest cardinality of a color set C, so that Alice has a winning strategy for the game played on D with C. A digraph is A-perfect if, for any induced subdigraph H of D, the A-game chromatic number of H equals the size of the largest symmetric clique of H. We characterize some basic classes of A-perfect digraphs, in particular all A-perfect semiorientations of paths and cycles. This gives us, as corollaries, similar results for other games, in particular concerning the digraph version of the usual game chromatic number.