On the number of spanning trees a planar graph can have

K. Buchin, A. Schulz

Research output: Chapter in Book/Report/Conference proceedingConference contributionAcademicpeer-review

27 Citations (Scopus)

Abstract

We prove that any planar graph on n vertices has less than O(5.2852^n) 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(2.7156^n). As a consequence of the latter the grid size needed to realize a 3d polytope with integer coordinates can be bounded by O(147.7 n^). Our observations imply improved upper bounds for related quantities: the number of cycle-free graphs in a planar graph is bounded by O(6.4884^n), the number of plane spanning trees on a set of n points in the plane is bounded by O(158.6^n), and the number of plane cycle-free graphs on a set of n points in the plane is bounded by O(194.7^n).
Original languageEnglish
Title of host publicationAlgorithms - ESA 2010 (18th Annual European Symposium, Liverpool, UK, September 6-8, 2010. Proceedings, Part I)
EditorsM. Berg, de, U. Meyer
Place of PublicationBerlin
PublisherSpringer
Pages110-121
ISBN (Print)978-3-642-15774-5
DOIs
Publication statusPublished - 2010

Publication series

NameLecture Notes in Computer Science
Volume6346
ISSN (Print)0302-9743

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