On the connectivity of unit distance graphs
Given a field \(K\subseteq R\) and an integer \(d>1\), the graph \(G(K^d)\) is defined whose vertices are elements of \(K^d\), with an edge between any two points at (Euclidean) distance 1. The author proves that \(G(K^2)\) is not connected while \(G(K^d)\) is connected for \(d\geq5\); \(G(K^4)\) is connected if and only if \(G(K^3)\) is connected. Also necessary and sufficient conditions for the connectedness of \(G(K^3)\) are given.
scientific article; zbMATH DE number 934041