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On the Number of Spanning Trees a Planar Graph Can Have

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Book cover Algorithms – ESA 2010 (ESA 2010)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 6346))

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Abstract

We prove that any planar graph on n vertices has less than O(5.2852n) spanning trees. Under the restriction that the planar graph is 3-connected and contains no triangle and no quadrilateral the number of its spanning trees is less than O(2.7156n). As a consequence of the latter the grid size needed to realize a 3d polytope with integer coordinates can be bounded by O(147.7n). Our observations imply improved upper bounds for related quantities: the number of cycle-free graphs in a planar graph is bounded by O(6.4884n), the number of plane spanning trees on a set of n points in the plane is bounded by O(158.6n), and the number of plane cycle-free graphs on a set of n points in the plane is bounded by O(194.7n).

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Author information

Authors and Affiliations

  1. Department of Mathematics and Computer Science, Technical University of Eindhoven,  

    Kevin Buchin

  2. Institut für Mathematsche Logik und Grundlagenforschung, Universität Münster,  

    André Schulz

Authors
  1. Kevin Buchin
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  2. André Schulz
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Editor information

Editors and Affiliations

  1. Department of Mathematics and Computing Science, TU Eindhoven, Eindhoven, The Netherlands

    Mark de Berg

  2. Institute for Computer Science, J.W. Goethe University, 60325, Frankfurt/Main, Germany

    Ulrich Meyer

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Buchin, K., Schulz, A. (2010). On the Number of Spanning Trees a Planar Graph Can Have. In: de Berg, M., Meyer, U. (eds) Algorithms – ESA 2010. ESA 2010. Lecture Notes in Computer Science, vol 6346. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-15775-2_10

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  • DOI: https://doi.org/10.1007/978-3-642-15775-2_10

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-15774-5

  • Online ISBN: 978-3-642-15775-2

  • eBook Packages: Computer ScienceComputer Science (R0)

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