Physical Sciences and Mathematics Commons^™

Connectedness Of Unit Distance Subgraphs Induced By Closed Convex Sets, Remie Janssen, Leonie Van Steijn

Theory and Applications of Graphs

The unit distance graph G¹_R^d is the infinite graph whose nodes are points in R^d, with an edge between two points if the Euclidean distance between these points is 1. The 2-dimensional version G¹_R2 of this graph is typically studied for its chromatic number, as in the Hadwiger-Nelson problem. However, other properties of unit distance graphs are rarely studied. Here, we consider the restriction of G¹_Rd to closed convex subsets X of R^d. We show that the graph G¹_Rd[X] is connected precisely when the radius of …