Generalized DP-Colorings of Graphs

By a graph we mean a finite undirected graph having multiple edges but no loops. Given a graph property $\mathcal{P}$, a $\mathcal{P}$-coloring of a graph $G$ with color set $C$ is a mapping $\text{f}:V(G)\to C$ such that for each color $c\in C$ the subgraph of $G$ induced by the color class ${\phi }^{-1}(c)$ belongs to $\mathcal{P}$. The $\mathcal{P}$-chromatic number $\chi (G:\mathcal{P})$ of $G$ is the least number $k$ for which $G$ admits an $\mathcal{P}$-coloring with a set of $k$-colors. This coloring concept dates back to the late 1960s and is commonly known as generalized coloring. In the 1980s the $\mathcal{P}$-choice number ${\chi }_{\mathrm{\ell }}(G:\mathcal{P})$ of $G$ was introduced and investigated by several authors. In 2018 \v{D}vor'ak and Postle introduced the DP-chromatic number as a natural extension of the choice number. They also remarked that this concept applies to any graph property. This motivated us to investigate the $\mathcal{P}$-DP-chromatic number $\chi_{\rm DP}(G:$\mathcal{P}$)$of$G$\mathrm{.}Wehave$\chi(G:$\mathcal{P}$)\leq \chi_\ell(G:$\mathcal{P}$)\leq \chi_{\rm DP}(G:$\mathcal{P}$)$. In this paper we show that various fundamental coloring results, in particular, the theorems of Brooks, of Gallai, and of Erd\H{o}s, Rubin and Taylor, have counterparts for the $\mathcal{P}$-DP-chromatic number. Furthermore, we provide a generalization of a result from 2000 about partition of graphs into a fixed number of induced subgraphs with bounded variable degeneracy due to Borodin, Kostochka, and Toft.