随机顺序模型下的FirstFit在线着色算法
FirstFit online coloring in the random order model
- Shenzhen University of Advanced Technology（深圳先进技术研究院）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文研究随机顺序模型下FirstFit在线着色算法，将其树图分析推广至稀疏图类，证明其在冠图上仅用O(1)颜色，明确密度不足以保证二分图的O(1)颜色，指出部分图类随机改进有限并提出开放问题。
AI中文摘要：
近期Frei等人与Bosek等人的研究完全确定了随机顺序模型中FirstFit在线着色算法在树图上的平均性能，结果显示其使用的颜色数为Θ(log n / log log n)，优于对抗模型下的Θ(log n)颜色数。本文针对稍更一般的图类给出若干进一步结果：首先，将其方法推广得到适用于稀疏图类的简单路径计数原理，该原理立即得出仙人掌图与均匀超树也存在类似改进；接着证明FirstFit在冠图上仅使用O(1)颜色，而冠图是对抗到达时需Θ(n)颜色的标准示例；进一步表明仅密度（即使是线性最小度）不足以保证二分图上也能使用O(1)颜色；最后确定单位区间图及一些高色数图等图类，随机到达仅能提供有限改进。文末列出若干开放问题。
英文摘要：
The average performance of FirstFit online coloring on trees in the random order model is completely determined in recent works of Frei et al. and Bosek et al., showing $Θ(\log n /\log\log n)$ number of colors, improving the $Θ(\log n)$ colors in the adversarial model. We provide a few further results on slightly more general graph classes. Firstly, we extend their method to obtain a simple path-counting principle for sparse graph classes, which immediately yields for example that cactus graphs and uniform hypertrees exhibit a similar improvement. We then show that FirstFit uses only $O(1)$ colors on crown graphs, a standard example where adversarial arrival forces $Θ(n)$ colors. We further show that density alone (even linear minimum degree) is insufficient to guarantee $O(1)$ colors even on bipartite graphs. Finally, we identify graph classes, including unit interval graphs and some graphs of high chromatic number, for which random arrival provides only limited improvement. We end with some open problems.
