Abstract
It is known that there are normal plane maps (NPMs) with minimum degree δ = 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 δ = 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 δ = 5 has no 4-stars of cyclic type \(\overrightarrow {\left( {5,6,6,5} \right)} \) 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 \(\overrightarrow {\left( {5,6,6,5} \right)} \)-free NPM with δ = 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 (\(\overrightarrow {\left( {5,6,6,5} \right)} \)-free NPM with δ = 5.
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References
Wernicke P., “Über den kartographischen Vierfarbensatz,” Math. Ann., 58, 413–426 (1904).
Franklin Ph., “The four colour problem,” Amer. J. Math., 44, 225–236 (1922).
Lebesgue H., “Quelques conséquences simples de la formule d’Euler,” J. Math. Pures Appl., 19, 27–43 (1940).
Jendrol’ S. and Madaras T., “On light subgraphs in plane graphs with minimum degree five,” Discuss. Math. Graph Theory, 16, 207–217 (1996).
Borodin O. V. and Woodall D. R., “Short cycles of low weight in normal plane maps with minimum degree 5,” Discuss. Math. Graph Theory, 18, No. 2, 159–164 (1998).
Borodin O. V. and Ivanova A. O., “Describing 4-stars at 5-vertices in normal plane maps with minimum degree 5,” Discrete Math., 313, No. 17, 1710–1714 (2013).
Van den Heuvel J. and McGuinness S., “Coloring the square of a planar graph,” J. Graph Theory, 42, 110–124 (2003).
Balogh J., Kochol M., Pluhár A., and Yu X., “Covering planar graphs with forests,” J. Comb. Theory Ser. B, 94, 147–158 (2005).
Harant J. and Jendrol’ S., “On the existence of specific stars in planar graphs,” Graphs Comb., 23, 529–543 (2007).
Borodin O. V. and Ivanova A. O., “Describing (d-2)-stars at d-vertices, d ≤ 5, in normal plane maps,” Discrete Math., 313, No. 17, 1700–1709 (2013).
Borodin O. V., Broersma H. J., Glebov A. N., and van den Heuvel J., “The structure of plane triangulations in terms of clusters and stars,” Diskret. Anal. Issled. Oper. Ser. 1, 8, No. 2, 15–39 (2001).
Borodin O. V., Broersma H. J., Glebov A. N., and van den Heuvel J., “Minimal degrees and chromatic numbers of squares of planar graphs,” Diskret. Anal. Issled. Oper. Ser. 1, 8, No. 4, 9–33 (2001).
Jendrol’ S. and Madaras T., “Note on an existence of small degree vertices with at most one big degree neighbour in planar graphs,” Tatra Mt. Math. Publ., 30, 149–153 (2005).
Borodin O. V., Ivanova A. O., and Jensen T. R., “5-stars of low weight in normal plane maps with minimum degree 5,” Discuss. Math. Graph Theory, 34, No. 3, 539–546 (2014).
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Novosibirsk; Yakutsk. Translated from Sibirskiĭ Matematicheskiĭ Zhurnal, Vol. 57, No. 3, pp. 596–602, May–June, 2016; DOI: 10.17377/smzh.2016.57.307.
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Borodin, O.V., Ivanova, A.O. Light and low 5-stars in normal plane maps with minimum degree 5. Sib Math J 57, 470–475 (2016). https://doi.org/10.1134/S0037446616030071
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DOI: https://doi.org/10.1134/S0037446616030071