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An Eberhard-like theorem for pentagons and heptagons | | Matt DeVos Agelos Georgakopoulos Bojan Mohar Robert Šámal
; | | Date: |
21 May 2009 | | Abstract: | Eberhard proved that for every sequence $(p_k), 3le kle r, k
e 5,7$ of
non-negative integers satisfying Euler’s formula $sum_{kge3} (6-k) p_k = 12$,
there are infinitely many values $p_6$ such that there exists a simple convex
polyhedron having precisely $p_k$ faces of length $k$ for every $kge3$, where
$p_k=0$ if $k>r$. In this paper we prove a similar statement when non-negative
integers $p_k$ are given for $3le kle r$, except for $k=5$ and $k=7$. 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 length $k$ for every
$kge3$. %, where $p_k=0$ if $k>r$. 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. | | Source: | arXiv, 0905.3504 | | Services: | Forum | Review | PDF | Favorites | |
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