发表机构
University of Illinois Urbana-Champaign（伊利诺伊大学厄巴纳-香槟分校）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
研究在DP - 染色背景下顶点荫度的轻微推广，通过此框架得出顶点荫度临界图和列表荫度临界图边数的下界，比Gallai界更好。
AI 中文摘要
多重图\(G\)的顶点荫度\(\mathrm{va}(G)\)是使得\(V(G)\)能够被划分为\(k\)个子集的最小数\(k\)，其中每个子集在\(G\)中诱导出一个无环子图。根据定义，若\(\mathrm{va}(G)= k\)，则色数\(\chi(G)\)满足\(k\leq \chi(G)\leq 2k\)。1976年Borodin以及1979年Bollobás和Manvel的基本结果暗示了\((2k - 1)\) - 临界图中边数的Gallai下界的类似结果。我们在DP - 染色的背景下考虑顶点荫度的一个轻微推广。利用这个框架，我们得到了对于顶点荫度临界图和列表荫度临界图的边数的下界，这些下界比Gallai界更好，以及在我们的DP - 设置中的类似界。
英文摘要
The vertex arboricity $\mathrm{va}(G)$ of a multigraph $G$ is the minimum number $k$ for which $V(G)$ can be partitioned into $k$ subsets, each of which induces an acyclic subgraph of $G$. By definition, if $\mathrm{va}(G)= k$, then the chromatic number, $χ(G)$, satisfies $k\leq χ(G)\leq 2k$. Fundamental results by Borodin from 1976 and Bollobás and Manvel from 1979 imply an analog of Gallai's lower bound on the number of edges in a $(2k-1)$-critical graph. We consider a slight generalization of vertex arboricity in the setting of DP-coloring. Using this framework, we derive lower bounds on the number of edges in graphs critical for vertex arboricity and for list arboricity that are better than Gallai's bound, along with similar bounds in our DP-setting.
