Journal ArticleDOI
Weight of faces in plane maps
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In this article, exact upper bounds for the minimum weight of minor faces in normal plane maps and 3-polytopes with specified maximum vertex degree were obtained for the case of 3 polytopes.Abstract:
Precise upper bounds are obtained for the minimum weight of minor faces in normal plane maps and 3-polytopes with specified maximum vertex degree.read more
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Journal Article
Stars and bunches in planar graphs. Part II: General planar graphs and colourings
TL;DR: The structure of plane graphs in terms of stars and bunches was studied in this article, where it was shown that a plane graph contains a $(d-1)$-star centred at a vertex of degree $d\leq5$ and the sum of the degrees of the vertices in the star is bounded.
Journal ArticleDOI
Describing 3-faces in normal plane maps with minimum degree 4
Oleg V. Borodin,Anna O. Ivanova +1 more
TL;DR: The following description of 3-faces in normal plane maps with minimum degree at least 4 holds for 3-polytopes in which every parameter is best possible and is attained independently of the others.
Journal ArticleDOI
Describing faces in plane triangulations
TL;DR: It is proved that in fact every plane triangulation contains a face with the vertex-degrees majorized by one of the following triples, where every parameter is tight.
Book ChapterDOI
A general framework for coloring problems: old results, new results, and open problems
TL;DR: A survey of the existing results, mainly based on and strongly biased by joint work of the author with several different groups of coauthors, include some new results, and discuss several open problems for each of the variants as mentioned in this paper.
Journal ArticleDOI
The vertex-face weight of edges in 3-polytopes
Oleg V. Borodin,Anna O. Ivanova +1 more
TL;DR: In this article, it was shown that each 3-polytope without pyramidal edges has an edge e with w(e) ≤ 18, which is the tight upper bound.
References
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Journal ArticleDOI
Triangulated 3-polytopes without faces of low weight
TL;DR: In this paper, precise upper bounds for the maximal length of a path consisting of 4-vertices in triangulated 3-polytopes were derived for the case where no two 4-vertex vertices are adjacent.
Journal ArticleDOI
Cyclic Degrees of 3-Polytopes
TL;DR: A precise upper bound is obtained for the minimum cyclic vertex degree in a 3- polytope with specified maximum face size, implying improvements to some known upper bounds for the cyclic chromatic numbers of 3-polytopes.
Journal ArticleDOI
Structure of neighborhoods of edges in planar graphs and simultaneous coloring of vertices, edges and faces
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Light subgraphs in planar graphs of minimum degree 4 and edge-degree 9
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