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A theorem on the structure of cell–decompositions of orientable 2–manifolds

Published online by Cambridge University Press:  26 February 2010

E. Jucovič  and
M. Trenkler
E. Jucovič
Affiliation:
P.J. Šafárik University, Košice, Czechoslovakia.
M. Trenkler
Affiliation:
P.J. Šafárik University, Košice, Czechoslovakia.
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Extract

For a cell compex M decomposing the closed orientable 2-manifold Pg of genus g let pi (M) and νj(M) denote the number of i-gonal cells (faces, countries) and j-valent vertices of the graph (= 1 – skeleton) of M, respectively. It will be supposed that i, j ≥ 3. From Euler's formula follows

and

is even, where e is the number of edges of M. As seen, the relation above does not impose restrictions on the numbers p4 (M), ν4(M). In an attempt to characterize the vectors {pi(M)}, {νi(M)}which partially determine the combinatorial structure of the cell-decompositions, the first step could involve answering the question: Given sequences p = (p3, p5, …), ν = (ν3, ν5, …) of non-negative integers satisfying conditions

and

does there exist a cell-decomposition M of Pg, for which pi (M) = pi, νj(M) = νj for all i, j ≠ 4? (If so, the sequences p, ν are called realizable in the sequel. M itself is a realization of p, ν. For brevity's sake we will often use the word map instead of cell-decomposition in the sequel.)

Type
Research Article
Information
Mathematika , Volume 20 , Issue 1 , June 1973 , pp. 63 - 82
Copyright
Copyright © University College London 1973

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References

1. Barnette, D.Jucovič, E.Trenkler, M., “Toroidal maps with prescribed types of vertices and faces”, Mathematika, 18 (1971), 82–90.CrossRefGoogle Scholar
2. Grünbaum, B., Convex polytopes (Wiley, New York, 1967).Google Scholar
3. Grünbaum, B., “Planar maps with prescribed types of vertices and faces”, Mathematica, 16 (1969), 28–36.Google Scholar
4. Jucovič, E. and Trenkler, M.: “On 4-valent graphs imbedded in orientable 2-manifolds”, Studia Scien. Math. Hungarica. (to appear).Google Scholar
5. Zaks, J.: “The analogue of Eberhard's theorem for 4-valent graphs on the torus”, Israel J. Math., 9 (1971), 299–305.Google Scholar