Geodesic
Describing Neighborhoods of 5-Vertices in 3-Polytopes with Minimum Degree 5 and Without Vertices of Degrees from 7 to 11
Discussiones Mathematicae. Graph Theory, Tome 38 (2018) no. 3, pp. 615-625

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In 1940, Lebesgue proved that every 3-polytope contains a 5-vertex for which the set of degrees of its neighbors is majorized by one of the following sequences: (6, 6, 7, 7, 7), (6, 6, 6, 7, 9), (6, 6, 6, 6, 11), (5, 6, 7, 7, 8), (5, 6, 6, 7, 12), (5, 6, 6, 8, 10), (5, 6, 6, 6, 17), (5, 5, 7, 7, 13), (5, 5, 7, 8, 10), (5, 5, 6, 7, 27), (5, 5, 6, 6, ∞), (5, 5, 6, 8, 15), (5, 5, 6, 9, 11), (5, 5, 5, 7, 41), (5, 5, 5, 8, 23), (5, 5, 5, 9, 17), (5, 5, 5, 10, 14), (5, 5, 5, 11, 13). In this paper we prove that every 3-polytope without vertices of degree from 7 to 11 contains a 5-vertex for which the set of degrees of its neighbors is majorized by one of the following sequences: (5, 5, 6, 6, ∞), (5, 6, 6, 6, 15), (6, 6, 6, 6, 6), where all parameters are tight.
Keywords: planar graph, structure properties, 3-polytope, neighborhood 
@article{DMGT_2018_38_3_a0,
     author = {Borodin, Oleg V. and Ivanova, Anna O. and Kazak, Olesya N.},
     title = {Describing {Neighborhoods} of {5-Vertices} in {3-Polytopes} with {Minimum} {Degree} 5 and {Without} {Vertices} of {Degrees} from 7 to 11},
     journal = {Discussiones Mathematicae. Graph Theory},
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Borodin, Oleg V.; Ivanova, Anna O.; Kazak, Olesya N. Describing Neighborhoods of 5-Vertices in 3-Polytopes with Minimum Degree 5 and Without Vertices of Degrees from 7 to 11. Discussiones Mathematicae. Graph Theory, Tome 38 (2018) no. 3, pp. 615-625. http://geodesic.mathdoc.fr/item/DMGT_2018_38_3_a0/