Cover and variable degeneracy

Abstract

Let $f$ be a nonnegative integer valued function on the vertex set of a graph. A graph is \textbf{strictly $f$-degenerate} if each nonempty subgraph $\Gamma$ has a vertex $v$ such that $\mathrm{deg}_{\Gamma}(v) < f(v)$. In this paper, we define a new concept, strictly $f$-degenerate transversal, which generalizes list coloring, signed coloring, DP-coloring, $L$-forested-coloring, and $(f_{1}, f_{2}, \dots, f_{s})$-partition. A \textbf{cover} of a graph $G$ is a graph $H$ with vertex set $V(H) = \bigcup_{v \in V(G)} X_{v}$, where $X_{v} = \{(v, 1), (v, 2), \dots, (v, s)\}$; the edge set $\mathscr{M} = \bigcup_{uv \in E(G)}\mathscr{M}_{uv}$, where $\mathscr{M}_{uv}$ is a matching between $X_{u}$ and $X_{v}$. A vertex set $R \subseteq V(H)$ is a \textbf{transversal} of $H$ if $|R \cap X_{v}| = 1$ for each $v \in V(G)$. A transversal $R$ is a \textbf{strictly $f$-degenerate transversal} if $H[R]$ is strictly $f$-degenerate. The main result of this paper is a degree type result, which generalizes Brooks' theorem, Gallai's theorem, degree-choosable result, signed degree-colorable result, and DP-degree-colorable result. We also give some structural results on critical graphs with respect to strictly $f$-degenerate transversal. Using these results, we can uniformly prove many new and known results. In the final section, we pose some open problems.