Two notions of unit distance graphs

A faithful (unit) distance graph in R-d is a graph whose set of vertices is a finite subset of the d-dimensional Euclidean space, where two vertices are adjacent if and only if the Euclidean distance between them is exactly 1. A (unit) distance graph in Rd is any subgraph of such a graph. In the first part of the paper we focus on the differences between these two classes of graphs. In particular, we show that for any fixed d the number of faithful distance graphs in Rd on n labelled vertices is 2((1+o(1))dn log2 n), and give a short proof of the known fact that the number of distance graphs in Rd on n labelled vertices is 2((1-1/left perpendiculard/2right perpendicular + o(1))n2/2.) We also study the behavior of several Ramsey-type quantities involving these graphs. In the second part of the paper we discuss the problem of determining the minimum possible number of edges of a graph which is not isomorphic to a faithful distance graph in R-d. (C) 2014 Elsevier Inc. All rights reserved.