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 (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.
