Colorful Results on Euclidean Distance Graphs and Their Chromatic Numbers

Abstract

In this work we study Euclidean distance graphs whose vertex set is the n-dimensional rational space. In particular, we deal with the chromatic numbers (and some related parameters) of such graphs when n = 2 or n = 3. A short history of the topic is given before we approach a few open problems related to the subject. Some of these questions are resolved, either partially or completely. For the problems able to resist our advances, methods are given that will hopefully lead to answers in future work.