In this paper we study fractional coloring from the angle of distributed computing. Fractional coloring is the linear relaxation of the classical notion of coloring, and has many applications, in particular in scheduling. It is known that for every real $\alpha>1$ and integer $\Delta$, a fractional coloring of total weight at most $\alpha(\Delta+1)$ can be obtained deterministically in a single round in graphs of maximum degree $\Delta$, in the LOCAL model of computation. However, a major issue of this result is that the output of each vertex has unbounded size. Here we prove that even if we impose the more realistic assumption that the output of each vertex has constant size, we can find fractional colourings with a weight arbitrarily close to known tight bounds for the fractional chromatic number in several cases of interest. Moreover, we improve on classical bounds on the chromatic number by considering the fractional chromatic number instead, without significantly increasing the output size and the round complexity of the existing algorithms.

翻译：在本文中, 我们从分布式计算的角度研究分数颜色。 分数颜色是传统色彩概念的线性松动, 并有许多应用, 特别是在列表中。 众所周知, 对于每个真实的 $alpha>1$和整数$\Delta$, 最多能以 $alpha (\ Delta+1) 来决定总重量的分数颜色。 在 LOCAL 计算模型中, 以最大度 $\ Delta$ 的图形来决定。 然而, 分数颜色颜色的分数是这个结果的一个主要问题, 即每个顶数的输出没有限制大小 。 我们在这里证明, 即使我们把每个顶数的输出量都设定为不变大小的更现实的假设, 我们也可以在几个感兴趣的案例中找到任意接近已知的分数色数字的紧紧界限的分数的分数色颜色。 此外, 我们通过考虑分数的染色体数字, 来改进染色的经典界限 。