Paintability of r-chromatic graphs

The online list coloring game is a two-player graph-coloring game played on a graph G as follows. On each turn, a Lister reveals a new color c at some subset $S\subseteq V\left(G\right)$ of uncolored vertices, and then a Painter chooses an independent subset of S to which to assign c. As the game is played, the revealed colors at each vertex $v\in V\left(G\right)$ form a color set $L\left(v\right)$, often called a list. The paintability of G measures the minimum value k for which Painter has a strategy to complete a coloring of G in such a way that $\left|L\right(v\left)\right|\le k$ for each vertex $v\in V\left(G\right)$. The paintability of a graph is an upper bound for its list chromatic number, or choosability.

The online list coloring game is a special case of the DP-painting game, which is defined similarly using the setting of DP-coloring. In the DP-painting game, the Lister reveals correspondence covers of a graph G rather than colors, and the Painter chooses independent subsets of these covers. The DP-painting game has a parameter known as DP-paintability which is analogous to paintability.

In this paper, we consider upper bounds for the paintability and DP-paintability of a graph G with large maximum degree Δ and chromatic number at most some fixed value r. We prove that the paintability of G is at most $(1-\frac{1}{4r+1})\Delta +2$ and that the DP-paintability of G is at most $\Delta -\Omega \left(\sqrt{\Delta \log \Delta }\right)$. We prove our first upper bound using Alon-Tarsi orientations, and we prove our second upper bound by considering the strict type-3 degeneracy parameter recently introduced by Zhou, Zhu, and Zhu.