A Generalization of Some Results on List Coloring and DP-Coloring

Let G be a graph and let fi,i∈{1,…,s},\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f_i, i \in \{1,\ldots ,s\},$$\end{document} be a function from V(G) to the set of nonnegative integers. In Sittitrai and Nakprasit (Analogue of DP-coloring on variable degeneracy and its applications, 2020), the concept of DP-F-coloring, a generalization of DP-coloring and variable degeneracy, was introduced. We use DP-F-coloring to define DPG-[k, t]-colorable graphs and modify the proofs in Liu et al. (Graphs Combin 35(3), 695–705, 2019), Sittitrai and Nakprasit (Bull Malays Math Sci Soc, 2019, https://doi.org/10.1007/s40840-019-00800-1), Thomassen (J Combin Theory Ser B 62, 180–181, 1994) to obtain more results on list coloring, DP-coloring, list-forested coloring, and variable degeneracy.