×

zbMATH — the first resource for mathematics

Joint extension of two theorems of Kotzig on 3-polytopes. (English) Zbl 0777.05050
By a 3-polytope is meant a planar map whose underlying graph is 3- connected. The weight of an edge is defined to be the sum of the degrees of its two ends. This paper provides that each 3-polytope has either an edge of weight at most 13 for which both incident faces are triangles, or an edge of weight at most 10 which is incident with a triangle, or else an edge of weight at most 8. All the bounds are shown to be sharp.

MSC:
05C10 Planar graphs; geometric and topological aspects of graph theory
05C15 Coloring of graphs and hypergraphs
PDF BibTeX Cite
Full Text: DOI
References:
[1] O. V. Borodin: Simultaneous coloring the vertices, edges, and faces of planar graphs,Met. Diskr. Anal., Novosibirsk 47 (1988), 21-27 (in Russian).
[2] O. V. Borodin: New structural properties of planar graphs with application in colorings,Proc. 33rd Int. Wiss. Kolloq. TH Ilmenau, Ilmenau, 1988, 159-162.
[3] O. V. Borodin: On the total coloring of planar graphs,J. Reine Angew. Math. 394 (1989), 180-185. · Zbl 0653.05029
[4] B. Gr?nbaum: New views on some old questions of combinatorial geometry,Teorie Combinatorie, Accademia Nacionale dei Lincei Roma, 1976, Vol. I, 452-468.
[5] A. Kotzig: Contribution to the theory of Eulerian polyhedra,Mat. ?as. 5 (1955), 233-237 (in Russian).
[6] A. Kotzig: From the theory of Euler’s polyhedrons,Mat. ?as. 13 (1963), 20-34 (in Russian). · Zbl 0134.19601
[7] J. Mitchem andH. Kronk: A seven-color theorem on the sphere,Discrete Math. 5 (1973), 253-260. · Zbl 0256.05106
[8] E. Steinitz: Polyeder und Raumeinleitungen,Enzyklop. Math. Wiss. 3 (1922), 1-139.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.