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

 

Rhombic embeddings of planar quad-graphs


Authors: Richard Kenyon and Jean-Marc Schlenker
Journal: Trans. Amer. Math. Soc. 357 (2005), 3443-3458
MSC (2000): Primary 52Cxx
Published electronically: May 10, 2004
MathSciNet review: 2146632
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Abstract | References | Similar Articles | Additional Information

Abstract: Given a finite or infinite planar graph all of whose faces have degree $4$, we study embeddings in the plane in which all edges have length $1$, that is, in which every face is a rhombus. We give a necessary and sufficient condition for the existence of such an embedding, as well as a description of the set of all such embeddings.

RÉSUMÉ. Etant donné un graphe planaire, fini ou infini, dont toutes les faces sont de degré $4$, on étudie ses plongements dans le plan dont toutes les arêtes sont de longueur $1$, c'est à dire dont toutes les faces sont des losanges. On donne une condition nécessaire et suffisante pour l'existence d'un tel plongement, et on décrit l'ensemble de ces plongements.


References [Enhancements On Off] (What's this?)

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Additional Information

Richard Kenyon
Affiliation: Laboratoire de Mathématiques, CNRS UMR 8628, Université Paris-Sud, 91405 Orsay, France

Jean-Marc Schlenker
Affiliation: Laboratoire Emile Picard, UMR CNRS 5580, Institut de Mathématiques, Université Paul Sabatier, 118 route de Narbonne, 31062 Toulouse Cedex 4, France
Email: schlenker@picard.ups-tlse.fr

DOI: http://dx.doi.org/10.1090/S0002-9947-04-03545-7
PII: S 0002-9947(04)03545-7
Received by editor(s): June 18, 2003
Received by editor(s) in revised form: September 15, 2003
Published electronically: May 10, 2004
Article copyright: © Copyright 2004 American Mathematical Society