We present three sublinear randomized algorithms for vertex-coloring of graphs with maximum degree $\Delta$. The first is a simple algorithm that extends the idea of Morris and Song to color graphs with maximum degree $\Delta$ using $\Delta+1$ colors. Combined with the greedy algorithm, it achieves an expected runtime of $O(n^{3/2}\sqrt{\log n})$ in the query model, improving on Assadi, Chen, and Khanna's algorithm by a $\sqrt{\log n}$ factor in expectation. When we allow quantum queries to the graph, we can accelerate the first algorithm using Grover's famous algorithm, resulting in a runtime of $\tilde{O}(n^{4/3})$ quantum queries. Finally, we introduce a quantum algorithm for $(1+\epsilon)\Delta$-coloring, achieving $O(\epsilon^{-1}n^{5/4}\log^{3/2}n)$ quantum queries, offering a polynomial improvement over the previous best bound by Morris and Song.

翻译：我们提出了三种针对最大度为$\Delta$的图进行顶点着色的亚线性随机算法。第一种是简单算法，它扩展了Morris和Song的思想，使用$\Delta+1$种颜色对最大度为$\Delta$的图进行着色。结合贪心算法，该算法在查询模型中的期望运行时间为$O(n^{3/2}\sqrt{\log n})$，在期望意义上比Assadi、Chen和Khanna的算法改进了$\sqrt{\log n}$因子。当我们允许对图进行量子查询时，可以利用Grover的著名算法加速第一种算法，从而得到$\tilde{O}(n^{4/3})$的量子查询运行时间。最后，我们提出了一种用于$(1+\epsilon)\Delta$着色的量子算法，实现了$O(\epsilon^{-1}n^{5/4}\log^{3/2}n)$的量子查询次数，相比Morris和Song先前的最佳界限提供了多项式级别的改进。