Joint extension of two theorems of Kotzig on 3-polytopes


The weight of an edge in a graph is the sum of the degrees of its end-vertices. It is proved that in each 3-polytope there exists 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 13, 10, and 8 are sharp and attained independently of each other.

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  1. [1]

    O. V. Borodin: Simultaneous coloring the vertices, edges, and faces of planar graphs,Met. Diskr. Anal., Novosibirsk 47 (1988), 21–27 (in Russian).

    Google Scholar 

  2. [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. [3]

    O. V. Borodin: On the total coloring of planar graphs,J. Reine Angew. Math. 394 (1989), 180–185.

    Google Scholar 

  4. [4]

    B. Grünbaum: New views on some old questions of combinatorial geometry,Teorie Combinatorie, Accademia Nacionale dei Lincei Roma, 1976, Vol. I, 452–468.

    Google Scholar 

  5. [5]

    A. Kotzig: Contribution to the theory of Eulerian polyhedra,Mat. čas. 5 (1955), 233–237 (in Russian).

    Google Scholar 

  6. [6]

    A. Kotzig: From the theory of Euler's polyhedrons,Mat. čas. 13 (1963), 20–34 (in Russian).

    Google Scholar 

  7. [7]

    J. Mitchem andH. Kronk: A seven-color theorem on the sphere,Discrete Math. 5 (1973), 253–260.

    Google Scholar 

  8. [8]

    E. Steinitz: Polyeder und Raumeinleitungen,Enzyklop. Math. Wiss. 3 (1922), 1–139.

    Google Scholar 

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Borodin, O. Joint extension of two theorems of Kotzig on 3-polytopes. Combinatorica 13, 121–125 (1993).

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AMS subject classification code (1991)

  • 05 C 10
  • 05 C 15