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Turán Type Results for Distance Graphs

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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 \( \lambda = \frac{2}{7} \).

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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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Authors and Affiliations

  1. Mathematics Faculty, Higher School of Economics, Moscow, Russia

    Lev E. Shabanov

  2. Department of Mathematical Statistics and Random Processes, Mechanics and Mathematics Faculty, Moscow State University, Moscow, Russia

    Andrei M. Raigorodskii

  3. Department of Discrete Mathematics and Laboratory of Advanced Combinatorics and Network Applications, Faculty of Innovations and High Technology, Moscow Institute of Physics and Technology, Moscow, Russia

    Andrei M. Raigorodskii

  4. Institute of Mathematics and Informatics, Buryat State University, Ulan-Ude, Russia

    Andrei M. Raigorodskii

Authors
  1. Lev E. Shabanov
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  2. Andrei M. Raigorodskii
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Correspondence to Andrei M. Raigorodskii.

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Editor in Charge: János Pach

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Shabanov, L.E., Raigorodskii, A.M. Turán Type Results for Distance Graphs. Discrete Comput Geom 56, 814–832 (2016). https://doi.org/10.1007/s00454-016-9817-z

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  • DOI: https://doi.org/10.1007/s00454-016-9817-z

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