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- W4297538796 abstract "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 $f:V(G)to C$ such that for each color $cin C$ the subgraph of $G$ induced by the color class $varphi^{-1}(c)$ 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 $chi_ell(G:mathcal{P})$ 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 $G$. We have $chi(G:mathcal{P})leq chi_ell(G:mathcal{P})leq chi_{rm DP}(G:mathcal{P})$. In this paper we show that various fundamental coloring results, in particular, the theorems of Brooks, of Gallai, and of ErdH{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." @default.
- W4297538796 created "2022-09-29" @default.
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- W4297538796 date "2019-08-01" @default.
- W4297538796 modified "2023-09-24" @default.
- W4297538796 title "Generalized DP-Colorings of Graphs" @default.
- W4297538796 doi "https://doi.org/10.48550/arxiv.1908.00282" @default.
- W4297538796 hasPublicationYear "2019" @default.
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