Two notions of unit distance graphs

Noga Alon, Andrey Kupavskii

Research output: Contribution to journalArticle

8 Scopus citations

Abstract

A faithful (unit) distance graph in Rd 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))dnlog2n, 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/⌊d/2⌋+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 Rd.

Original languageEnglish (US)
Pages (from-to)1-17
Number of pages17
JournalJournal of Combinatorial Theory. Series A
Volume125
Issue number1
DOIs
StatePublished - Jul 1 2014
Externally publishedYes

All Science Journal Classification (ASJC) codes

  • Theoretical Computer Science
  • Discrete Mathematics and Combinatorics
  • Computational Theory and Mathematics

Keywords

  • Graph dimension
  • Graph representation
  • Unit distance graph

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