The incidence game chromatic number of (a,d)(a,d)-decomposable graphs

The incidence coloring game has been introduced in Andres (2009) [2] as a variation of the ordinary coloring game. The incidence game chromatic number ιg(G)ιg(G) of a graph G is the minimum number of colors for which Alice has a winning strategy when playing the incidence coloring game on G.In Charpentier and Sopena (2013) [7], we proved that ιg(G)≤⌊3Δ(G)−a2⌋+8a−1 for every graph G with arboricity at most a  . In this paper, we extend our previous result to (a,d)(a,d)-decomposable graphs – that is graphs whose set of edges can be partitioned into two parts, one inducing a graph with arboricity at most a, the other inducing a graph with maximum degree at most d   – and prove that ιg(G)≤⌊3Δ(G)−a2⌋+8a+3d−1 for every (a,d)(a,d)-decomposable graph G.