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Discrete & Computational Geometry

, Volume 56, Issue 3, pp 814–832 | Cite as

Turán Type Results for Distance Graphs

  • Lev E. Shabanov
  • Andrei M. RaigorodskiiEmail author
Article

Abstract

The classical Turán theorem determines the minimum number of edges in a graph on n vertices with independence number \(\alpha \). We consider unit-distance graphs on the Euclidean plane, i.e., graphs \( G = (V,E) \) with \( V \subset {\mathbb {R}}^2 \) and \( E = \{\{\mathbf{x}, \mathbf{y}\}: |\mathbf{x}-\mathbf{y}| = 1\} \), and show that the minimum number of edges in a unit-distance graph on n vertices with independence number \( \alpha \leqslant \lambda n \), \( \lambda \in [\frac{1}{4}, \frac{2}{7}] \), is bounded from below by the quantity \( \frac{19 - 50 \lambda }{3} n \), which is several times larger than the general Turán bound and is tight at least for \( \lambda = \frac{2}{7} \).

Keywords

Turán theorem Independence number Distance graphs 

Mathematics Subject Classification

05C35 52C10 

Notes

Acknowledgments

This work is done under the financial support of the following grants: the Grant 15-01-00350 of Russian Foundation for Basic Research, the Grant NSh-2964.2014.1 supporting leading scientific schools of Russia.

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Copyright information

© Springer Science+Business Media New York 2016

Authors and Affiliations

  1. 1.Mathematics FacultyHigher School of EconomicsMoscowRussia
  2. 2.Department of Mathematical Statistics and Random Processes, Mechanics and Mathematics FacultyMoscow State UniversityMoscowRussia
  3. 3.Department of Discrete Mathematics and Laboratory of Advanced Combinatorics and Network Applications, Faculty of Innovations and High TechnologyMoscow Institute of Physics and TechnologyMoscowRussia
  4. 4.Institute of Mathematics and InformaticsBuryat State UniversityUlan-UdeRussia

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