Problem
Research questions and friction points this paper is trying to address.
FirstFit
online coloring
random order model
graph classes
color number
FirstFit online coloring in the random order model
🤖 AI Summary
研究使用FirstFit在线着色方法在随机顺序模型下对不同图类进行着色，发现该方法在稀疏图类中表现更优，并探讨了密度对结果的影响。
📝 Abstract
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.
Innovation
Methods, ideas, or system contributions that make the work stand out.
FirstFit online coloring
sparse graph classes
path-counting principle
cactus graphs
uniform hypertrees
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