Directed defective asymmetric graph coloring games

Abstract

We consider the following 2-person game which is played with an (initially uncolored) digraph D , a finite color set C , and nonnegative integers a , b , and d . Alternately, player I colors a vertices and player II colors b vertices with colors from C . Whenever a player colors a vertex v , all in-arcs ( w , v ) that do not come from a vertex w previously colored with the same color as v are deleted. For each color i the defect digraph D i is the digraph induced by the vertices of color i at a certain state of the game. The main rule the players have to respect is that at every time for any color i the digraph D i has maximum total degree of at most d . The game ends if no vertex can be colored any more according to this rule. Player I wins if D is completely colored at the end of the game, otherwise player II wins. The smallest cardinality of a color set C with which player I has a winning strategy for the game is called d - r e l a x e d ( a , b ) - g a m e c h r o m a t i c n u m b e r of D . This parameter generalizes several variants of Bodlaender’s game chromatic number. We determine the tight (resp., nearly tight) upper bound ⌊ b d + 1 ⌋ + 2 (resp., ⌊ b d + 1 ⌋ + 3 ) for the d -relaxed ( a , b ) -game chromatic number of orientations of forests (resp., undirected forests) for any d and a ≥ b ≥ 1 . Furthermore we prove that these numbers cannot be bounded in case a < b .