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.
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).
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
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\paper Light and low $5$-stars in normal plane maps with minimum degree~$5$
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Linking options:
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This publication is cited in the following 14 articles:
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
O. V. Borodin, A. O. Ivanova, “Soft $3$-stars in sparse plane graphs”, Sib. elektron. matem. izv., 17 (2020), 1863–1868
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
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
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
Ya. Li, M. Rao, T. Wang, “Minor stars in plane graphs with minimum degree five”, Discret Appl. Math., 257 (2019), 233–242
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
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
O. V. Borodin, A. O. Ivanova, “Light 3-stars in sparse plane graphs”, Sib. elektron. matem. izv., 15 (2018), 1344–1352
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
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
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
Oleg V. Borodin, Anna O. Ivanova, AIP Conference Proceedings, 1903, 2017, 030051
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