RUI: Graph Coloring Parameters: Their Interplay with Number Theory Problems and Applications to Broadcast Communications

我们（PI和合作者）研究了两个主要主题。 第一个是关于距离图的分数和圆染色以及它们与两个数论问题的密切联系，即缺失差的整数序列的密度和“孤独跑步者猜想”中涉及的参数。我们应用这些连接，以加强着色问题和数论问题的研究。此外，我们还研究了Kneser图的圆色数，重点研究了约化Kneser图的圆色数。 第二个主要的主题是由信道分配问题。我们研究距离二标记，通过使用圆形距离二标记作为一个有效的工具，我们扩展到多级距离标记（或“无线电标记”）的研究，更广泛的实际应用。图的着色问题吸引了研究者们长达世纪之久。这部分是由于各种着色参数之间的迷人联系以及与其他领域问题的联系，部分是由于它们丰富的实际应用。PI和她的合作者研究了几个图着色问题，包括分数着色，圆形着色和着色的通道分配问题的动机。 分数染色和圆染色都可以看作是传统顶点染色的推广，在过去的二十年里得到了广泛的研究。 研究这些着色参数与数论问题之间的联系，不仅可以为数论问题的研究提供新的视角，而且可以促进这一领域的知识发展。 同样，对信道分配变化的研究为实际应用提供了模型，并拓宽了目前对各种类型的距离标记（着色）的研究。 该项目的主要合作者是台湾国立中山大学的Xuding Zhu。

We (the PI and collaborators) investigate two major topics. The first is about fractional and circular coloring for distance graphs and their close connections to two number theory problems, namely, the density of integral sequences with missing differences and the parameter involved in the "lonely runner conjecture". We apply these connections to enhance the study of both coloring problems and problems in number theory. In addition, we investigate the circular chromatic number for Kneser graphs, with an emphasis on the reduced Kneser graphs. The second major topic is motivated by the channel assignment problem. We study distance two labeling, by using circular distance two labeling as an effective tool, and we extend the study to multi-level distance labeling (or "radio labeling"), for broader practical applications. Graph coloring problems have attracted researchers for more than a century. This is partially due to fascinating connections among various coloring parameters and connections to problems in other fields, and partially due to their abundant practical applications. The PI and her collaborators investigate several graph coloring problems including fractional coloring, circular coloring, and colorings motivated by the channel assignment problem. Both fractional coloring and circular coloring can be regarded as generalizations of the conventional vertex coloring, and have been studied extensively in the past two decades. Research on connections among these coloring parameters and problems in number theory not only provides new insight into, but also advances the knowledge in these fields. Likewise, research on variations of the channel assignment provides models to practical applications, and broadens the current study on various types of distance labeling (coloring). A major collaborator to the project is Xuding Zhu, National Sun Yet-Sen University, Taiwan.