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Sibirsk. Mat. Zh., 2016, Volume 57, Number 3, Pages 596–602 (Mi smj2765)  

This article is cited in 5 scientific papers (total in 5 papers)

Light and low $5$-stars in normal plane maps with minimum degree $5$

O. V. Borodina, A. O. Ivanovab

a Sobolev Institute of Mathematics, Siberian Branch of the Russian Academy of Sciences, Novosibirsk, Russia
b Ammosov North-Eastern Federal University, Yakutsk, Russia

Abstract: It is known that there are normal plane maps (NPMs) with minimum degree $\delta=5$ such that the minimum degree-sum $w(S_5)$ of $5$-stars at $5$-vertices is arbitrarily large. The height of a $5$-star is the maximum degree of its vertices. Given an NPM with $\delta=5$, by $h(S_5)$ we denote the minimum height of a $5$-stars at $5$-vertices in it.
Lebesgue showed in 1940 that if an NPM with $\delta=5$ has no $4$-stars of cyclic type $(\overrightarrow{5,6,6,5})$ centered at $5$-vertices, then $w(S_5)<68$ and $h(S_5)<41$. Recently, Borodin, Ivanova, and Jensen lowered these bounds to $55$ and $28$, respectively, and gave a construction of a $(\overrightarrow{5,6,6,5})$-free NPM with $\delta=5$ having $w(S_5)=48$ and $h(S_5)=20$.
In this paper, we prove that $w(S_5)<51$ and $h(S_5)<23$ for each $(\overrightarrow{5,6,6,5})$-free NPM with $\delta=5$.

Keywords: graph, plane map, weight, light subgraph, height, low subgraph.

Funding Agency Grant Number
Russian Foundation for Basic Research 16-01-00499
15-01-05867
12-01-98510
Ministry of Education and Science of the Russian Federation НШ-1939.2014.1
The first author was supported by the Russian Foundation for Basic Research (Grants 16-01-00499 and 15-01-05867) and the State Maintenance Program for the Leading Scientific Schools of the Russian Federation (Grant NSh-1939.2014.1). The second author worked within the governmental task “Organization of Scientific Research” and supported by the Russian Foundation for Basic Research (Grant 15-01-05867).


DOI: https://doi.org/10.17377/smzh.2016.57.307

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English version:
Siberian Mathematical Journal, 2016, 57:3, 470–475

Bibliographic databases:

Document Type: Article
UDC: 519.17
Received: 17.09.2015

Citation: O. V. Borodin, A. O. Ivanova, “Light and low $5$-stars in normal plane maps with minimum degree $5$”, Sibirsk. Mat. Zh., 57:3 (2016), 596–602; Siberian Math. J., 57:3 (2016), 470–475

Citation in format AMSBIB
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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. O. V. Borodin, A. O. Ivanova, “Light neighborhoods of $5$-vertices in $3$-polytopes with minimum degree $5$”, Sib. elektron. matem. izv., 13 (2016), 584–591  mathnet  crossref
    2. O. V. Borodin, A. O. Ivanova, D. V. Nikiforov, “Low minor $5$-stars in $3$-polytopes with minimum degree $5$ and no $6$-vertices”, Discrete Math., 340:7 (2017), 1612–1616  crossref  mathscinet  zmath  isi  scopus
    3. O. V. Borodin, A. O. Ivanova, D. V. Nikiforov, “Low and light $5$-stars in $3$-polytopes with minimum degree $5$ and restrictions on the degrees of major vertices”, Siberian Math. J., 58:4 (2017), 600–605  mathnet  crossref  crossref  isi  elib  elib
    4. O. V. Borodin, A. O. Ivanova, D. V. Nikiforov, “Describing neighborhoods of $5$-vertices in a class of $3$-polytopes with minimum degree $5$”, Siberian Math. J., 59:1 (2018), 43–49  mathnet  crossref  crossref  isi  elib
    5. O. V. Borodin, A. O. Ivanova, “Light 3-stars in sparse plane graphs”, Sib. elektron. matem. izv., 15 (2018), 1344–1352  mathnet  crossref
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