AI 中文总结
本文证实了高围长d-退化图分数色数的上界猜想，给出高效随机算法，还建立了更强形式的下界，揭示均匀附着模型无Erdős-Rényi图中的算法间隙。
AI 中文摘要
Martinsson和Steiner近期证明，任何d-退化无三角形图G的分数色数χ_f(G)满足χ_f(G)=O(d/log d)，并进一步猜想其主项紧常数为1+o(1)。本文证实了该猜想对围长至少为5的图成立。我们的证明是构造性的：它给出一个高效随机算法，该算法以高概率计算此类图中权重至多为(1+o(1))d/log d的分数着色。此外，我们以更强形式建立了他们的下界猜想：对任意常数g≥4，存在围长至少为g的d-退化图，其χ_f(G)≥(1-o(1))d/log d，该下界通过分析基于均匀附着模型的随机图得到。值得注意的是，我们的结果揭示该模型不存在Erdős-Rényi图中发现的典型计算复杂性障碍，而后者对该问题存在猜想的因子2算法间隙。
英文摘要
Martinsson and Steiner recently proved that the fractional chromatic number of any $d$-degenerate triangle-free graph $G$ satisfies $χ_f(G) = O\left(\frac{d}{\log d}\right)$. They further conjectured a sharp leading constant $1 + o(1)$. In this paper, we confirm their upper bound conjecture for graphs having girth at least $5$. Our proof is constructive: it gives an efficient randomized algorithm that, with high probability, computes a fractional coloring of weight at most $(1 + o(1))\frac{d}{\log d}$ in such graphs. Furthermore, we establish their conjectured lower bound in a stronger form: for any constant $g \ge 4$, there exist $d$-degenerate graphs having girth at least $g$ with $χ_f(G) \ge (1 - o(1))\frac{d}{\log d}$. This lower bound is achieved by analyzing a random graph based on the uniform attachment model. Notably, our results reveal that this model lacks the typical computational complexity barriers found in Erdős-Rényi graphs, where there is a conjectured factor-$2$ algorithmic gap for this problem.
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