Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: 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 $\f:V(G)\to C$ such that for each
color $c\in C$ 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 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.

14
The categories of this paper are: math-ph
8
The categories of this paper are: hep-ex
19
The categories of this paper are: eess
31
The categories of this paper are: ao-sci
28
The categories of this paper are: alg-geom
1
The categories of this paper are: cs
4
The categories of this paper are: cond-mat
11
The categories of this paper are: hep-lat
21
The categories of this paper are: adap-org
24
The categories of this paper are: dg-ga
15
The categories of this paper are: nucl-ex
32
The categories of this paper are: atom-ph
23
The categories of this paper are: solv-int
22
The categories of this paper are: chao-dyn
10
The categories of this paper are: q-bio
27
The categories of this paper are: comp-gas