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 $i_g(G)$ 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 %$\lceil 3/2 \Delta(G)\rceil \le$i_g(G) \le 2\Delta(G) + 4k - 2$for every$k$-degenerate graph$G$.$a(G)$of a graph$G$is the minimum number of forests into which its set of edges can be partitioned.$G$is$k$-degenerate, then$a(G) \le k \le 2a(G) - 1$. We show in this paper that$i_g(G) \le \lfloor\frac{3\Delta(G) - a(G)}{2}\rfloor + 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) \le k$for every$k$-degenerate graph$G$. Since there exists graphs with$i_g(G) \ge \lceil\frac{3\Delta(G)}{2}\rceil$, the multiplicative constant of our bound is best possible.