Two notions of unit distance graphs

TL;DR

This paper compares faithful and general unit distance graphs in Euclidean spaces, providing asymptotic counts and exploring Ramsey properties, while also investigating minimal edges for non-isomorphic faithful distance graphs.

Contribution

It establishes asymptotic formulas for the number of faithful and general distance graphs in fixed dimensions and examines related Ramsey-type properties and minimal edge counts.

Findings

Number of faithful distance graphs in d7^d on n vertices is 2^{(1+o(1)) d n \u2217 log_2 n}

Number of distance graphs in d7^d on n vertices is 2^{(1-1/floor d/2 f) n^2/2}

Analysis of minimal edges in graphs not isomorphic to faithful distance graphs

Abstract

A {\em faithful (unit) distance graph} in $\mathbb{R}^d$ 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 {\em (unit) distance graph} in $\mathbb{R}^d$ 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 $\mathbb{R}^d$ on $n$ labelled vertices is $2^{(1+o(1)) d n \log_2 n}$, and give a short proof of the known fact that the number of distance graphs in $\mathbb{R}^d$ on $n$ labelled vertices is $2^{(1-1/\lfloor d/2 \rfloor +o(1))n^2/2}$. We also study the behavior of several Ramsey-type quantities involving these graphs. % and high-girth graphs from these classes. 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 $\mathbb R^d$.