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

The incidence coloring game has been introduced in [S.D. Andres, The incidence game chromatic number, Discrete Appl. Math. 157 (2009), 1980– 1987] as a variation of the ordinary coloring game. The incidence game chromatic number 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 [C. Charpentier and É. Sopena, Incidence coloring game and arboricity of graphs, Proc. IWOCA’2013, Lecture Notes Comput. Sci. 8288 (2013), 106–114], we proved that ιg(G) ≤ ⌊ 2 ⌋ + 8a − 1 for every graph G with arboricity at most a. In this paper, we extend our previous result to (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) ≤ ⌊ 2 ⌋+ 8a+ 3d− 1 for every (a, d)-decomposable graph G.