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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

An analogue of the Descartes-Euler formula for infinite graphs and Higuchi's conjecture


Authors: Matt DeVos and Bojan Mohar
Journal: Trans. Amer. Math. Soc. 359 (2007), 3287-3300
MSC (2000): Primary 05C10
Published electronically: February 21, 2007
MathSciNet review: 2299456
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Abstract: Let $ \mathcal{R}$ be a connected 2-manifold without boundary obtained from a (possibly infinite) collection of polygons by identifying them along edges of equal length. Let $ V$ be the set of vertices, and for every $ v \in V$, let $ \kappa(v)$ denote the (Gaussian) curvature of $ v$: $ 2 \pi$ minus the sum of incident polygon angles. Descartes showed that $ \sum_{v \in V} \kappa(v) = 4 \pi$ whenever $ \mathcal{R}$ may be realized as the surface of a convex polytope in $ \mathbb{R}^3$. More generally, if $ \mathcal{R}$ is made of finitely many polygons, Euler's formula is equivalent to the equation $ \sum_{v \in V} \kappa(v) = 2 \pi \chi(\mathcal{R})$ where $ \chi(\mathcal{R})$ is the Euler characteristic of $ \mathcal{R}$. Our main theorem shows that whenever $ \sum_{v \in V : \kappa(v) < 0} \kappa(v)$ converges and there is a positive lower bound on the distance between any pair of vertices in $ \mathcal{R}$, there exists a compact closed 2-manifold $ \mathcal{S}$ and an integer $ t$ so that $ \mathcal{R}$ is homeomorphic to $ \mathcal{S}$ minus $ t$ points, and further $ \sum_{v \in V} \kappa(v) \le 2 \pi \chi(\mathcal{S}) - 2 \pi t$.

In the special case when every polygon is regular of side length one and $ \kappa(v) > 0$ for every vertex $ v$, we apply our main theorem to deduce that $ \mathcal{R}$ is made of finitely many polygons and is homeomorphic to either the 2-sphere or to the projective plane. Further, we show that unless $ \mathcal{R}$ is a prism, antiprism, or the projective planar analogue of one of these that $ \vert V\vert \le 3444$. This resolves a recent conjecture of Higuchi.


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Additional Information

Matt DeVos
Affiliation: Department of Mathematics, Simon Fraser University, Burnaby, British Columbia, V5A 1S6, Canada
Email: mdevos@sfu.ca

Bojan Mohar
Affiliation: Department of Mathematics, University of Ljubljana, 1000 Ljubljana, Slovenia
Address at time of publication: Department of Mathematics, Simon Fraser University, Burnaby, British Columbia, V5A 1S6, Canada
Email: bojan.mohar@fmf.uni-lj.si

DOI: http://dx.doi.org/10.1090/S0002-9947-07-04125-6
PII: S 0002-9947(07)04125-6
Received by editor(s): July 2, 2004
Received by editor(s) in revised form: May 11, 2005
Published electronically: February 21, 2007
Additional Notes: The first author was supported in part by the SLO-USA Grant BI-US/04-05/36 and by the Slovenian grant L1–5014.
The second author was supported in part by the Ministry of Education, Science and Sport of Slovenia, Research Program P1–0297 and Research Project J1–6150.
Article copyright: © Copyright 2007 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.