Paintability of r-chromatic graphs

The paintability of a graph is a coloring parameter defined in terms of an online list coloring game. In this paper we ask, what is the paintability of a graph G of maximum degree and chromatic number r? By considering the Alon-Tarsi number of G, we prove that the paintability of G is at most (1 - 14r+1) + 2. We also consider the DP-paintability of G, which is defined in terms of an online DP-coloring game. By considering the strict type 3 degeneracy parameter recently introduced by Zhou, Zhu, and Zhu, we show that when r is fixed and is sufficiently large, the DP-paintability of G is at most - ().