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 Diskr. Mat., 2016, Volume 28, Issue 4, Pages 139–149 (Mi dm1398)

On the number of maximal independent sets in complete $q$-ary trees

D. S. Taletskiia, D. S. Malyshevb

a Lobachevski State University of Nizhni Novgorod
b State University – Higher School of Economics in Nizhnii Novgorod

Abstract: The paper is concerned with the asymptotic behaviour of the number $\operatorname{mi}(T_{q,n})$ of maximal independent sets in a complete $q$-ary tree of height $n$. For some constants $\alpha_2$ and $\beta_2$ the asymptotic formula $\operatorname{mi}(T_{2,n})\thicksim \alpha_2\cdot (\beta_2)^{2^n}$ is shown to hold as $n\to\infty$. It is also proved that $\operatorname{mi}(T_{q,3k})\thicksim \alpha^{(1)}_q\cdot(\beta_q)^{q^{3k}},\operatorname{mi}(T_{q,3k+1})\thicksim \alpha^{(2)}_q\cdot(\beta_q)^{q^{3k+1}},\operatorname{mi}(T_{q,3k+2})\thicksim \alpha^{(3)}_q\cdot(\beta_q)^{q^{3k+2}}$ as $k\to \infty$ for any sufficiently large $q$, some three pairwise distinct constants $\alpha^{(1)}_q,\alpha^{(2)}_q,\alpha^{(3)}_q$ and a constant $b_q$.

Keywords: maximal independent set, complete $q$-ary tree.

 Funding Agency Grant Number Russian Foundation for Basic Research 16-31-60008_ìîë_à_äê This research was carried out with the financial support of the Russian Foundation for Basic Research (grant no. 16-31-60008-mol_a_dk) and the Laboratory of algorithms and analysis of network structures at the National Research University “Higher School of Economics”, Nizhny Novgorod Branch.

DOI: https://doi.org/10.4213/dm1398

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English version:
Discrete Mathematics and Applications, 2017, 27:5, 311–318

Bibliographic databases:

UDC: 519.172.1

Citation: D. S. Taletskii, D. S. Malyshev, “On the number of maximal independent sets in complete $q$-ary trees”, Diskr. Mat., 28:4 (2016), 139–149; Discrete Math. Appl., 27:5 (2017), 311–318

Citation in format AMSBIB
\Bibitem{TalMal16}
\by D.~S.~Taletskii, D.~S.~Malyshev
\paper On the number of maximal independent sets in complete $q$-ary trees
\jour Diskr. Mat.
\yr 2016
\vol 28
\issue 4
\pages 139--149
\mathnet{http://mi.mathnet.ru/dm1398}
\crossref{https://doi.org/10.4213/dm1398}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=3699327}
\elib{http://elibrary.ru/item.asp?id=28119098}
\transl
\jour Discrete Math. Appl.
\yr 2017
\vol 27
\issue 5
\pages 311--318
\crossref{https://doi.org/10.1515/dma-2017-0032}
\scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-85031778325}

• http://mi.mathnet.ru/eng/dm1398
• https://doi.org/10.4213/dm1398
• http://mi.mathnet.ru/eng/dm/v28/i4/p139

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This publication is cited in the following articles:
1. D. S. Taletskii, “O proizvodyaschikh funktsiyakh i predelnykh teoremakh, svyazannykh s maksimalnymi nezavisimymi mnozhestvami v grafakh-reshetkakh”, Zhurnal SVMO, 19:2 (2017), 105–116
2. D. S. Taletskii, “O svoistvakh resheniya rekurrentnogo uravneniya, perechislyayuschego maksimalnye nezavisimye mnozhestva v polnykh derevyakh”, Zhurnal SVMO, 20:1 (2018), 46–54
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