Print Email Facebook Twitter Connectedness of Unit Distance Subgraphs Induced by Closed Convex Sets Title Connectedness of Unit Distance Subgraphs Induced by Closed Convex Sets Author Janssen, R. (TU Delft Discrete Mathematics and Optimization) van Steijn, Leonie (Universiteit Leiden) Date 2022 Abstract The unit distance graph G1Rd is the infinite graph whose nodes are points in Rd, with an edge between two points if the Euclidean distance between these points is 1. The 2-dimensional version G1R2 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 G1Rd to closed convex subsets X of Rd. We show that the graph G1Rd[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. To reference this document use: http://resolver.tudelft.nl/uuid:0742e42f-43cf-41c9-8b27-98d4fb1734e5 DOI https://doi.org/10.20429/tag.2022.090102 ISSN 2470-9859 Source Theory and Applications of Graphs, 9 (1) Part of collection Institutional Repository Document type journal article Rights © 2022 R. Janssen, Leonie van Steijn Files PDF Unit_Distance_Graphs_of_C ... x_Sets.pdf 1.2 MB Close viewer /islandora/object/uuid:0742e42f-43cf-41c9-8b27-98d4fb1734e5/datastream/OBJ/view