Incidence Coloring Game and Arboricity of Graphs

Abstract

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 $i_g(G) \le 2\varDelta(G) + 4k - 2$ for every k-degenerate graph G. We show in this paper that $i_g(G) \le \lfloor\frac{3\varDelta(G) - a(G)}{2}\rfloor + 8a(G) - 2$ for every graph G, where a(G) stands for the arboricity of G, thus improving the bound given by Andres since a(G) ≤ k for every k-degenerate graph G. Since there exists graphs with $i_g(G) \ge \lceil\frac{3\varDelta(G)}{2}\rceil$, the multiplicative constant of our bound is best possible.