Graph coloring has a wide range of real world applications, such as in the operations research, communication network, computational biology and compiler optimization fields. In our recent work [1], we propose a divide-and-conquer approach for graph coloring, called VColor. Such an approach has three generic subroutines. (i) Graph partition subroutine: VColor partitions a graph G into a vertex cut partition (VP), which comprises a vertex cut component (VCC) and small non-overlapping connected components (CCs). (ii) Component coloring subroutine: VColor colors the VCC and the CCs by efficient algorithms. (iii) Color combination subroutine: VColor combines the local colors by exploiting the maximum matchings of color combination bigraphs (CCBs). VColor has revealed some major bottlenecks of efficiency in these subroutines. Therefore, in this paper, we propose VColor*, an approach which addresses these efficiency bottlenecks without using more colors both theoretically and experimentally. The technical novelties of this paper are the following. (i) We propose the augmented VP to index the crossing edges of the VCC and the CCs and propose an optimized CCB construction algorithm. (ii) For sparse CCs, we propose using a greedy coloring algorithm that is of polynomial time complexity in the worst case, while preserving the approximation ratio. (iii) We propose a distributed graph coloring algorithm. Our extensive experimental evaluation on real-world graphs confirms the efficiency of VColor*. In particular, VColor* is 20X and 50X faster than VColor and uses the same number of colors with VColor on the Pokec and PA datasets, respectively. VColor* also significantly outperforms the state-of-the-art graph coloring methods.

图形着色在现实世界中具有广泛的应用，例如在运筹学，通信网络，计算生物学和编译器优化领域。在我们最近的工作[1]中，我们提出了一种用于图形着色的分治法，称为VColor。这种方法具有三个通用子例程。（i）图分区子例程：VColor将图G划分为一个顶点切割分区（VP），该分区包括一个顶点切割组件（VCC）和小的非重叠连接组件（CC）。（ii）组件着色子例程：VColor通过高效的算法为VCC和CC着色。（iii）配色子程序：VColor通过利用颜色组合图（CCB）的最大匹配来组合局部颜色。VColor在这些子例程中揭示了一些主要的效率瓶颈。因此，在本文中，我们提出了VColor *，这是一种解决这些效率瓶颈的方法，在理论上和实验上都无需使用更多的颜色。本文的技术新颖性如下。（i）我们建议增加副总裁索引VCC和CC的交叉边缘，并提出一种优化的CCB构造算法。（ii）对于稀疏CC，我们建议使用在最坏情况下具有多项式时间复杂度的贪婪着色算法，同时保留近似比率。（iii）我们提出了一种分布式图形着色算法。我们对真实图形的广泛实验评估证实了VColor *的效率。特别是，VColor *比VColor快20倍和50倍，并且分别在Pokec和PA数据集上使用与VColor相同数量的颜色。VColor *还明显优于最新的图形着色方法。