## 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 language | English (US) |
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Pages (from-to) | 1-17 |

Number of pages | 17 |

Journal | Journal of Combinatorial Theory. Series A |

Volume | 125 |

Issue number | 1 |

DOIs | |

State | Published - Jul 2014 |

Externally published | Yes |

## 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