We introduce the fractional version of oriented coloring and initiate its study. We prove some basic results and study the parameter for directed cycles and sparse planar graphs. In particular, we show that for every $\epsilon > 0$, there exists an integer $g_{\epsilon} \geq 12$ such that any oriented planar graph having girth at least $g_{\epsilon}$ has fractional oriented chromatic number at most $4+\epsilon$. Whereas, it is known that there exists an oriented planar graph having girth at least $g_{\epsilon}$ with oriented chromatic number equal to $5$. We also study the fractional oriented chromatic number of directed cycles and provide its exact value. Interestingly, the result depends on the prime divisors of the length of the directed cycle.

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关于定向着色的分数版本

我们介绍了定向着色的分数版本并开始了它的研究。我们证明了一些基本结果并研究了有向环和稀疏平面图的参数。特别地，我们证明对于每一个 $\epsilon > 0$，存在一个整数 $g_{\epsilon} \geq 12$ 使得任何周长至少为 $g_{\epsilon}$ 的有向平面图都具有分数取向的色度数最多 $4+\epsilon$。然而，已知存在周长至少为$g_{\epsilon}$且定向色数等于$5$的定向平面图。我们还研究了定向循环的分数定向色数并提供了它的确切值。有趣的是，结果取决于有向循环长度的主要因数。