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• W2956603337 abstract "Let $f$ be a nonnegative integer valued function on the vertex set of a graph. A graph is {bf 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 {bf 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 {bf transversal} of $H$ if $|R cap X_{v}| = 1$ for each $v in V(G)$. A transversal $R$ is a {bf 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. Similar to Borodin, Kostochka and Toft's variable degeneracy, this degree type result is also self-strengthening. 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." @default.
• W2956603337 created "2019-07-23" @default.
• W2956603337 creator A5000556109 @default.
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• W2956603337 date "2019-07-13" @default.
• W2956603337 modified "2023-09-27" @default.
• W2956603337 title "Cover and variable degeneracy." @default.
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