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 $(\frac{1}{2} + o(1))\cdot \ln n \,/\, \ln\ln n$ 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：本文研究一种图着色算法，该算法按均匀随机顺序处理顶点，并采用首次适应策略为其分配颜色。我们证明，对于任意包含$n$个顶点的森林，该算法期望使用的颜色数不超过$(\frac{1}{2} + o(1))\cdot \ln n \,/\, \ln\ln n$种。同时，我们构造了一族森林，表明该上界是紧的。