Incidence coloring game and arboricity of graphs

TL;DR

This paper investigates the incidence coloring game on graphs, establishing an improved upper bound for the incidence game chromatic number in terms of maximum degree and arboricity, which tightens previous bounds.

Contribution

The paper introduces a new upper bound for the incidence game chromatic number based on arboricity, refining earlier results and demonstrating the bound's tightness with existing examples.

Findings

New upper bound: $i_g(G) \,\le\, \lfloor\frac{3\Delta(G) - a(G)}{2}\rfloor + 8a(G) - 2$

Bound is tight up to a constant factor, matching known lower bounds

Improves previous bounds for graphs with given arboricity

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 %$\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.