No-three-in-line-in-3D

Attila Pór, David R. Wood

Research output: Chapter in Book/Report/Conference proceedingConference PaperResearchpeer-review

3 Citations (Scopus)

Abstract

The no-three-in-line problem, introduced by Dudeney in 1917, asks for the maximum number of points in the n × n grid with no three points collinear. In 1951, Erdös proved that the answer is θ(n). We consider the analogous three-dimensional problem, and prove that the maximum number of points in the n × n × n grid with no three collinear is θ(n 2). This result is generalised by the notion of a 3D drawing of a graph. Here each vertex is represented by a distinct gridpoint in ℤ3, such that the line-segment representing each edge does not intersect any vertex, except for its own endpoints. Note that edges may cross. A 3D drawing of a complete graph Kn is nothing more than a set of n gridpoints with no three collinear. A slight generalisation of our first result is that the minimum volume for a 3D drawing of Kn is θ(n 3/2). This compares favourably to θ(n3) when edges are not allowed to cross. Generalising the construction for Kn, we prove that every k-colourable graph on n vertices has a 3D drawing with O(n√k) volume. For the k-partite Turán graph, we prove a lower bound of Ω((kn)3/4).

Original languageEnglish
Title of host publicationGraph Drawing
Subtitle of host publication12th International Symposium, GD 2004 New York, NY, USA, September 29-October 2, 2004 Revised Selected Papers
EditorsJános Pach
Place of PublicationBerlin Germany
PublisherSpringer
Pages395-402
Number of pages8
ISBN (Print)3540245286
DOIs
Publication statusPublished - 1 Dec 2004
Externally publishedYes
EventGraph Drawing 2004 - New York, United States of America
Duration: 29 Sep 20042 Oct 2004
Conference number: 12th
https://link.springer.com/book/10.1007/b105810 (Proceedings)

Publication series

NameLecture Notes in Computer Science
PublisherSpringer
Volume3383
ISSN (Print)0302-9743

Conference

ConferenceGraph Drawing 2004
Abbreviated titleGD 2004
CountryUnited States of America
CityNew York
Period29/09/042/10/04
Internet address

Cite this