Coloring unit-disk graphs efficiently is an important problem in the global and distributed setting, with applications in radio channel assignment problems when the communication relies on omni-directional antennas of the same power. In this context it is important to bound not only the complexity of the coloring algorithms, but also the number of colors used. In this paper, we consider two natural distributed settings. In the location-aware setting (when nodes know their coordinates in the plane), we give a constant time distributed algorithm coloring any unit-disk graph $G$ with at most $4\omega(G)$ colors, where $\omega(G)$ is the clique number of $G$. This improves upon a classical 3-approximation algorithm for this problem, for all unit-disk graphs whose chromatic number significantly exceeds their clique number. When nodes do not know their coordinates in the plane, we give a distributed algorithm in the LOCAL model that colors every unit-disk graph $G$ with at most $5.68\omega(G)$ colors in $O(\log^3 \log n)$ rounds. Moreover, when $\omega(G)=O(1)$, the algorithm runs in $O(\log^* n)$ rounds. This algorithm is based on a study of the local structure of unit-disk graphs, which is of independent interest. We conjecture that every unit-disk graph $G$ has average degree at most $4\omega(G)$, which would imply the existence of a $O(\log n)$ round algorithm coloring any unit-disk graph $G$ with (approximately) $4\omega(G)$ colors in the LOCAL model. We provide partial results towards this conjecture using Fourier-analytical tools.

翻译：调色单位- disk 图形效率是全球和分布式设置中的一个重要问题, 当通信依赖于同一电力的全向天线时, 无线电频道分配应用程序中的应用程序有问题。 在此情况下, 不仅必须约束颜色算法的复杂性, 而且还要约束所使用的颜色数量。 在本文中, 我们考虑到两个自然分布的设置。 当节点在平面上知道它们的坐标时, 我们给出一个固定时间分配的算法, 以任何单位- disk 图形( G$, 最多为 4\ omega( G) 美元, 其中$\ omga( G) 是 美元。 对于所有单位- disk 图表, 其色数大大超过其圆形数。 当节点在平面上不知道它们的坐标时, 我们给每个单位- disk 图形( $美元, 最多为 $. 68( G) 美元, 美元, 也就是以美元 美元 美元 的直径/ G 亚 亚 轨道结构中, 以美元表示每个单位 美元 美元 美元 美元 。