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An Eberhard-like theorem for pentagons and heptagons
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DeVos, Matt, Georgakopoulos, Agelos, Mohar, Bojan and Šámal, Robert (2010) An Eberhard-like theorem for pentagons and heptagons. Discrete & Computational Geometry, Vol.44 (No.4). pp. 931-945. doi:10.1007/s00454-010-9264-1 ISSN 0179-5376.
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Official URL: http://dx.doi.org/10.1007/s00454-010-9264-1
Abstract
Eberhard proved that for every sequence (p k ), 3≤k≤r, k≠6, of nonnegative integers satisfying Euler’s formula ∑ k≥3(6−k)p k =12, there are infinitely many values p 6 such that there exists a simple convex polyhedron having precisely p k faces of size k for every k≥3, where p k =0 if k>r. In this paper we prove a similar statement when nonnegative integers p k are given for 3≤k≤r, except for k=5 and k=7 (but including p 6). We prove that there are infinitely many values p 5,p 7 such that there exists a simple convex polyhedron having precisely p k faces of size k for every k≥3. We derive an extension to arbitrary closed surfaces, yielding maps of arbitrarily high face-width. Our proof suggests a general method for obtaining results of this kind.
Item Type: | Journal Article | ||||
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Divisions: | Faculty of Science, Engineering and Medicine > Science > Mathematics | ||||
Journal or Publication Title: | Discrete & Computational Geometry | ||||
Publisher: | Springer New York LLC | ||||
ISSN: | 0179-5376 | ||||
Official Date: | 2010 | ||||
Dates: |
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Volume: | Vol.44 | ||||
Number: | No.4 | ||||
Page Range: | pp. 931-945 | ||||
DOI: | 10.1007/s00454-010-9264-1 | ||||
Status: | Peer Reviewed | ||||
Publication Status: | Published | ||||
Access rights to Published version: | Restricted or Subscription Access |
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