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Polytopality and Cartesian products of graphs

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Abstract

We study the question of polytopality of graphs: when is a given graph the graph of a polytope? We first review the known necessary conditions for a graph to be polytopal, and we present three families of graphs which satisfy all these conditions, but which nonetheless are not graphs of polytopes.

Our main contribution concerns the polytopality of Cartesian products of non-polytopal graphs. On the one hand, we show that products of simple polytopes are the only simple polytopes whose graph is a product. On the other hand, we provide a general method to construct (non-simple) polytopal products whose factors are not polytopal.

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Authors and Affiliations

  1. Departament de Matemàtica Aplicada II, Universitat Politècnica de Catalunya, Barcelona, Spain

    Julian Pfeifle

  2. Laboratoire d’Informatique de l’École Polytechnique, École Polytechnique, 91128, Palaiseau Cedex, France

    Vincent Pilaud

  3. Departamento de Matemáticas Estadística y Computación, Universidad de Cantabria, Santander, Spain

    Francisco Santos

Authors
  1. Julian Pfeifle
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  2. Vincent Pilaud
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  3. Francisco Santos
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Corresponding author

Correspondence to Julian Pfeifle.

Additional information

Julian Pfeifle was partially supported by MICINN grants MTM2008-03020 and MTM2009-07242, AGAUR grant 2009 SGR 1040, and MICINN-ESF EUROCORES programme EuroGIGA — ComPoSe IP04 — Project EUI-EURC-2011-4306.

Vincent Pilaud was partially supported by MICINN grant MTM2008-04699-C03-02.

Francisco Santos was partially supported by MICINN under grants MTM2008-04699-C03-02 and CSD2006-00032 (i-MATH) and by MICINN-ESF EUROCORES programme EuroGIGA — ComPoSe IP04 — Project EUI-EURC-2011-4306.

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Pfeifle, J., Pilaud, V. & Santos, F. Polytopality and Cartesian products of graphs. Isr. J. Math. 192, 121–141 (2012). https://doi.org/10.1007/s11856-012-0049-5

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  • DOI: https://doi.org/10.1007/s11856-012-0049-5

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