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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
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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