It is well known that a random subgraph of the complete graph $K_n$ has chromatic number $\Theta(n/\log n)$ w.h.p. Boris Bukh asked whether the same holds for a random subgraph of any $n$-chromatic graph, at least in expectation. In this paper it is shown that for every graph, whose fractional chromatic number is at least $n$, the fractional chromatic number of its random subgraph is at least $n/(8\log_2(4n))$ with probability more than $1-\frac{1}{2n}$. This gives the affirmative answer for a strengthening of Bukh's question for the fractional chromatic number.

中文翻译：

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随机子图的分数色数

众所周知，完全图 $K_n$ 的随机子图具有色数 $\Theta(n/\log n)$ whp Boris Bukh 问是否同样适用于任何 $n$-色图的随机子图，至少在预期中。本文证明，对于每个图，其分数色数至少为$n$，其随机子图的分数色数至少为$n/(8\log_2(4n))$，概率大于$1 -\frac{1}{2n}$. 这为加强布赫关于分数色数的问题给出了肯定的答案。