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Zap. Nauchn. Sem. POMI, 2017, Volume 464, Pages 132–168 (Mi znsl6526)  

This article is cited in 3 scientific papers (total in 3 papers)

Turán type results for distance graphs in infinitesimal plane layer

L. E. Shabanov

Department of Innovations and High Technology, Moscow Institute of Physics and Technology, Dolgoprudnyi, Russia

Abstract: In this paper we obtain the lower bound of number of edges in a unit distance graph $\Gamma$ in an infinitesimal plane layer $\mathbb R^2\times[0,\varepsilon]^d$ which compares number of edges $e(\Gamma)$, number of vertices $\nu(\Gamma)$ and independence number $\alpha(\Gamma)$. Our bound $e(\Gamma)\ge\frac{19\nu\Gamma)-50\alpha(\Gamma)}3$ is generalizing of previous bound for distance graphs in plane and a strong upgrade of Turán's bound when $\frac15\le\frac{\alpha(\Gamma)}{\nu(\Gamma)}\le\frac27$.

Key words and phrases: distance graph, independence number, Turán type bounds.

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English version:
Journal of Mathematical Sciences (New York), 2019, 236:5, 554–578

UDC: 519.173
Received: 03.11.2017

Citation: L. E. Shabanov, “Turán type results for distance graphs in infinitesimal plane layer”, Combinatorics and graph theory. Part IX, Zap. Nauchn. Sem. POMI, 464, POMI, St. Petersburg, 2017, 132–168; J. Math. Sci. (N. Y.), 236:5 (2019), 554–578

Citation in format AMSBIB
\Bibitem{Sha17}
\by L.~E.~Shabanov
\paper Tur\'an type results for distance graphs in infinitesimal plane layer
\inbook Combinatorics and graph theory. Part~IX
\serial Zap. Nauchn. Sem. POMI
\yr 2017
\vol 464
\pages 132--168
\publ POMI
\publaddr St.~Petersburg
\mathnet{http://mi.mathnet.ru/znsl6526}
\transl
\jour J. Math. Sci. (N. Y.)
\yr 2019
\vol 236
\issue 5
\pages 554--578
\crossref{https://doi.org/10.1007/s10958-018-4133-1}


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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. Ph. A. Pushnyakov, “The Number of Edges in Induced Subgraphs of Some Distance Graphs”, Math. Notes, 105:4 (2019), 582–591  mathnet  crossref  crossref  isi  elib
    2. K. D. Kovalenko, A. M. Raigorodskii, “Systems of Representatives”, Math. Notes, 106:3 (2019), 372–377  mathnet  crossref  crossref  isi  elib
    3. Ph. A. Pushnyakov, A. M. Raigorodskii, “Estimate of the Number of Edges in Special Subgraphs of a Distance Graph”, Math. Notes, 107:2 (2020), 322–332  mathnet  crossref  crossref  isi  elib
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