d-退化局部稀疏图的分数DP染色
Fractional DP-colorings of $d$-degenerate locally sparse graphs
- 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总结：
本文将分数DP染色的上界从二部图推广到d-退化无三角形图，通过局部稀疏排序得到一般性上界，并构造例子证明其渐近最优，同时改进了若干图类的分数DP色数界。
AI中文摘要：
Bernshteyn、Kostochka和Zhu（2020）引入了分数DP染色的概念，该概念同时推广了分数染色和分数列表染色。在他们若干基础性结果中，他们证明了每个d-退化二部图G满足χ_f^DP ≤ (1 + o(1))d/log d，并且该界是最优的——这与普通分数染色形成鲜明对比。在本文中，我们将此上界推广到所有d-退化无三角形图，证明了χ_f^DP ≤ (4 + o(1))d/log d。这推广了Martinsson和Steiner（2025）关于普通分数染色的近期结果。我们将此结果作为关于局部稀疏图排序的更一般上界的推论。具体而言，一个d-退化图G是左k-局部稀疏的，如果它允许一个退化排序，使得对于每个顶点v，由其后邻诱导的子图至多包含k条边。我们证明，如果一个d-退化图G是左d^2/f-局部稀疏的，则χ_f^DP(G) ≤ (8 + o(1))d/log f。这立即给出普通分数色数χ_f(G)的相同上界，改进了先前已知界的前导常数。此外，我们建立了该结果在前导常数意义下的渐近锐性。对于任意1 ≪ f ≤ d^2，我们构造了左d^2/f-局部稀疏且满足χ_f(G) ≥ (1 - o(1))d/log f的d-退化图。最后，作为我们主定理的应用，我们获得了d-退化K_{1,t,t}-自由图以及最大度为Δ的K_{t,t,t}-自由图的分数DP色数的改进上界。值得注意的是，这些界即使在普通分数染色的设定中也改进了现有结果。
英文摘要：
Bernshteyn, Kostochka, and Zhu (2020) introduced the notion of fractional DP-coloring, which generalizes both fractional coloring and fractional list coloring. Among several foundational results, they proved that every $d$-degenerate bipartite graph $G$ satisfies $χ_f^{\mathrm{DP}} \le (1 + o(1))\frac{d}{\log d}$, and that this bound is optimal---a stark contrast to ordinary fractional coloring. In this paper, we extend this upper bound to all $d$-degenerate triangle-free graphs, proving that $χ_f^{\mathrm{DP}} \le (4 + o(1))\frac{d}{\log d}$. This generalizes a recent result of Martinsson and Steiner (2025) for ordinary fractional coloring. We derive this result as a corollary of a more general upper bound concerning locally sparse graph orderings. Specifically, a $d$-degenerate graph $G$ is left $k$-locally-sparse if it admits a degeneracy ordering in which, for every vertex $v$, the subgraph induced by its back-neighbors contains at most $k$ edges. We show that if a $d$-degenerate graph $G$ is left $\frac{d^2}{f}$-locally-sparse, then \[ χ_f^{\mathrm{DP}}(G) \le (8 + o(1))\frac{d}{\log f}. \] This immediately yields an identical upper bound on the ordinary fractional chromatic number $χ_f(G)$, improving upon the leading constants of previously known bounds. Additionally, we establish the asymptotic sharpness of this result up to the leading constant. For any $1 \ll f \le d^2$, we construct $d$-degenerate graphs that are left $\frac{d^2}{f}$-locally-sparse and satisfy $χ_f(G) \ge (1 - o(1))\frac{d}{\log f}$. Finally, as applications of our main theorem, we obtain improved upper bounds on the fractional DP-chromatic number of $d$-degenerate $K_{1,t,t}$-free graphs, as well as $K_{t,t,t}$-free graphs with maximum degree $Δ$. Notably, these bounds improve upon existing results even in the setting of ordinary fractional coloring.
