We consider a graph coloring algorithm that processes vertices in order taken uniformly at random and assigns colors to them using First-Fit strategy. We show that this algorithm uses, in expectation, at most $(1 + o(1))\cdot \ln n \,/\, \ln\ln n$ different colors to color any forest with $n$ vertices. We also construct a family of forests that shows that this bound is best possible.

翻译：我们研究一种图着色算法，该算法以均匀随机顺序处理顶点，并采用首适应策略为其分配颜色。我们证明，对于任意具有 $n$ 个顶点的森林，该算法在期望意义上至多使用 $(1 + o(1))\cdot \ln n \,/\, \ln\ln n$ 种不同颜色。我们还构造了一个森林族，表明该界是最优的。