Connectedness of Unit Distance Subgraphs Induced by Closed Convex Sets

Authors

Remie Janssen, Delft University of TechnologyFollow
Leonie van Steijn, Leiden UniversityFollow

Publication Date

February 2022

Abstract

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 r(X) of X is equal to 0, or when r(X) ≥ 1 and the affine dimension of X is at least 2. For hyperrectangles, we give bounds for the graph diameter in the critical case that the radius is exactly 1.

Creative Commons License

This work is licensed under a Creative Commons Attribution 4.0 License.

Recommended Citation

Janssen, Remie and van Steijn, Leonie (2022) "Connectedness of Unit Distance Subgraphs Induced by Closed Convex Sets," Theory and Applications of Graphs: Vol. 9: Iss. 1, Article 2.
DOI: 10.20429/tag.2022.090102
Available at: https://digitalcommons.georgiasouthern.edu/tag/vol9/iss1/2