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Discussiones Mathematicae Graph Theory 22(1) (2002) 39-50
DOI: https://doi.org/10.7151/dmgt.1157

SOME NEWS ABOUT OBLIQUE GRAPHS

Andrey A. Dobrynin  and  Leonid S. Melnikov  Sobolev Institute of Mathematics Siberian Branch of the Russian Academy of Science Novosibirsk 630090, Russia e-mail: dobr@math.nsc.ru e-mail: omeln@math.nsc.ru

Jens Schreyer  and  Hansjoachim Walther Technische Universität Ilmenau Germany e-mail: jens.schreyer@tu-ilmenau.de e-mail: hansjoachim.walther@tu-ilmenau.de

Abstract

A k-gon α of a polyhedral graph G(V,E,F) is of type ⟨b₁,b₂,…,b_k ⟩ if the vertices incident with α in cyclic order have degrees b₁,b₂,…,b_k and ⟨b₁,b₂,…,b_k ⟩ is the lexicographic minimum of all such sequences available for α. A polyhedral graph G is oblique if it has no two faces of the same type. Among others it is shown that an oblique graph contains vertices of degree 3.

References

[1]	O. Borodin, Structural properties of planar maps with the minimum degree 5, Math. Nachr. 158 (1992) 109-117, doi: 10.1002/mana.19921580108.
[2]	B. Grünbaum and C.J. Shephard, Spherical tilings with transitivity properties, in: Geometrie (Springer-Verlag, 1982) 65-98.
[3]	M. Voigt and H. Walther, Polyhedral graphs with restricted number of faces of the same type, Preprint No. M22/99, Technical University Ilmenau (submitted to Discr. Math.).
[4]	H. Walther, Polyhedral graphs with extreme numbers of types of faces, Preprint No. M13/99, Technical University Ilmenau (submitted to Appl. Discr. Math.).

Received 29 January 2001
Revised 1 August 2001