Note on induced subgraphs of the unit distance graph En

Abstract

The unit distance graph En is the graph whose vertices are the points in Euclidean n-space, and two vertices are adjacent if and only if the distance between them is 1. We prove that for any n there is a finite bipartite graph which cannot be embedded in En as an induced subgraph and that every finite graph with maximum degree d can be embedded in EN, N=(d3 -d)/2, as an induced subgraph. © 1989 Springer-Verlag New York Inc.