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 $S \subseteq V(G)$ 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 $v \in V(G)$ form a color set $L(v)$, 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 $|L(v)| \leq k$ for each vertex $v \in V(G)$. 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 $\Delta$ and chromatic number at most some fixed value $r$. We prove that the paintability of $G$ is at most $\left(1 - \frac{1}{4r+1} \right ) \Delta + 2$ and that the DP-paintability of $G$ is at most $\Delta - \Omega( \sqrt{\Delta \log \Delta})$. 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.