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Sibirskii Matematicheskii Zhurnal, 2016, Volume 57, Number 3, Pages 596–602
DOI: https://doi.org/10.17377/smzh.2016.57.307
(Mi smj2765)
 

This article is cited in 14 scientific papers (total in 14 papers)

Light and low 55-stars in normal plane maps with minimum degree 55

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
References:
Abstract: It is known that there are normal plane maps (NPMs) with minimum degree δ=5δ=5 such that the minimum degree-sum w(S5)w(S5) of 55-stars at 55-vertices is arbitrarily large. The height of a 55-star is the maximum degree of its vertices. Given an NPM with δ=5δ=5, by h(S5)h(S5) we denote the minimum height of a 55-stars at 55-vertices in it.
Lebesgue showed in 1940 that if an NPM with δ=5δ=5 has no 44-stars of cyclic type (5,6,6,5)(−−−−5,6,6,5) centered at 55-vertices, then w(S5)<68w(S5)<68 and h(S5)<41h(S5)<41. Recently, Borodin, Ivanova, and Jensen lowered these bounds to 5555 and 2828, respectively, and gave a construction of a (5,6,6,5)(−−−−5,6,6,5)-free NPM with δ=5δ=5 having w(S5)=48w(S5)=48 and h(S5)=20h(S5)=20.
In this paper, we prove that w(S5)<51w(S5)<51 and h(S5)<23h(S5)<23 for each (5,6,6,5)(−−−−5,6,6,5)-free NPM with δ=5δ=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).
Received: 17.09.2015
English version:
Siberian Mathematical Journal, 2016, Volume 57, Issue 3, Pages 470–475
DOI: https://doi.org/10.1134/S0037446616030071
Bibliographic databases:
Document Type: Article
UDC: 519.17
Language: Russian
Citation: O. V. Borodin, A. O. Ivanova, “Light and low 55-stars in normal plane maps with minimum degree 55”, Sibirsk. Mat. Zh., 57:3 (2016), 596–602; Siberian Math. J., 57:3 (2016), 470–475
Citation in format AMSBIB
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Linking options:
  • https://www.mathnet.ru/eng/smj2765
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  • This publication is cited in the following 14 articles:
    1. O.V. Borodin, A.O. Ivanova, “Almost all about light neighborhoods of 5-vertices in 3-polytopes with minimum degree 5”, Discrete Mathematics, 345:2 (2022), 112678  crossref
    2. O. V. Borodin, A. O. Ivanova, “Soft $3$-stars in sparse plane graphs”, Sib. elektron. matem. izv., 17 (2020), 1863–1868  mathnet  crossref
    3. O. V. Borodin, A. O. Ivanova, E. I. Vasil'eva, “Light minor 5-stars in 3-polytopes with minimum degree 5 and no 6-vertices”, Discuss. Math. Graph Theory, 40:4 (2020), 985–994  crossref  mathscinet  zmath  isi  scopus
    4. O. V. Borodin, M. A. Bykov, A. O. Ivanova, “Low 5-stars at 5-vertices in 3-polytopes with minimum degree 5 and no vertices of degree from 7 to 9”, Discuss. Math. Graph Theory, 40:4 (2020), 1025–1033  crossref  mathscinet  zmath  isi  scopus
    5. O. V. Borodin, A. O. Ivanova, “Light minor $5$-stars in $3$-polytopes with minimum degree $5$”, Siberian Math. J., 60:2 (2019), 272–278  mathnet  crossref  crossref  isi  elib
    6. Ya. Li, M. Rao, T. Wang, “Minor stars in plane graphs with minimum degree five”, Discret Appl. Math., 257 (2019), 233–242  crossref  mathscinet  zmath  isi  scopus
    7. O.V. Borodin, A.O. Ivanova, O.N. Kazak, “Describing the neighborhoods of 5-vertices in 3-polytopes with minimum degree 5 and no vertices of degree from 6 to 8”, Discrete Mathematics, 342:8 (2019), 2439  crossref
    8. 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
    9. O. V. Borodin, A. O. Ivanova, “Light 3-stars in sparse plane graphs”, Sib. elektron. matem. izv., 15 (2018), 1344–1352  mathnet  crossref
    10. 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
    11. 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
    12. O.V. Borodin, A.O. Ivanova, “On light neighborhoods of 5-vertices in 3-polytopes with minimum degree 5”, Discrete Mathematics, 340:9 (2017), 2234  crossref
    13. Oleg V. Borodin, Anna O. Ivanova, AIP Conference Proceedings, 1903, 2017, 030051  crossref
    14. 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
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