  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 $G$. %The arboricity $a(G)$ of a graph $G$ is the minimum
number of forests into which its set of edges can be partitioned. %If $G$ 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.
