Refined weight of edges in normal plane maps |
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Authors: | TsCh-D Batueva OV Borodin MA Bykov AO Ivanova ON Kazak DV Nikiforov |
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Institution: | 1. Sobolev Institute of Mathematics, Novosibirsk 630090, Russia;2. Novosibirsk State University, Novosibirsk 630090, Russia;3. Ammosov North-Eastern Federal University, Yakutsk, 677000, Russia |
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Abstract: | The weight of an edge in a normal plane map (NPM) is the degree-sum of its end-vertices. An edge is of type if and . In 1940, Lebesgue proved that every NPM has an edge of one of the types , , or , where 7 and 6 are best possible. In 1955, Kotzig proved that every 3-connected planar graph has an edge with , which bound is sharp. Borodin (1989), answering Erd?s’ question, proved that every NPM has either a -edge, or -edge, or -edge.A vertex is simplicial if it is completely surrounded by 3-faces. In 2010, Ferencová and Madaras conjectured (in different terms) that every 3-polytope without simplicial 3-vertices has an edge with . Recently, we confirmed this conjecture by proving that every NPM has either a simplicial 3-vertex adjacent to a vertex of degree at most 10, or an edge of types , , or .By a -vertex we mean a -vertex incident with precisely triangular faces. The purpose of our paper is to prove that every NPM has an edge of one of the following types: , , , , , , or , where all bounds are best possible. In particular, this implies that the bounds in , , and can be attained only at NPMs having a simplicial 3-, 4-, or 5-vertex, respectively. |
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Keywords: | Planar graph Plane map Structure properties 3-polytope Weight |
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