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 i_g(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 i_g(G) ≤ 2Δ(G) + 4k ? 2 for every k-degenerate graph G.We show in this paper that i_g(G) ≤ [(3Δ(G)?a(G))/2)]+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 i_g(G) ≥ [(3Δ(G))/2], the multiplicative constant of our bound is best possible.
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