Incidence Coloring Game and Arboricity of Graphs

An incidence of a graph G is a pair (v, e) where v is a vertex of G and e an edge incident to v. Two incidences (v, e) and (w, f) are adjacent whenever v = w, or e = f , or vw = e or f . The incidence coloring game [S.D. Andres, The incidence game chromatic number, Discrete Appl. Math. 157 (2009), 1980–1987] is a variation of the ordinary coloring game where the two players, Alice and Bob, alternately color the incidences of a graph, using a given number of colors, in such a way that adjacent incidences get distinct colors. If the whole graph is colored then Alice wins the game otherwise Bob wins the game. The incidence game chromatic number ig(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. Andres proved that ig(G) ≤ 2∆(G) + 4k − 2 for every k-degenerate graph G. We show in this paper that ig(G) ≤ b 2 c + 8a(G) − 2 for every graph G, where a(G) stands for the arboricity of G, thus improving the bound given by Andres since a(G) ≤ k for every k-degenerate graph G. Since there exists graphs with ig(G) ≥ d 3∆(G) 2 e, the multiplicative constant of our bound is best possible.