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(2.7156n ). 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). © 2010 Springer-Verlag.
CITATION STYLE
Buchin, K., & Schulz, A. (2010). On the number of spanning trees a planar graph can have. In Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) (Vol. 6346 LNCS, pp. 110–121). Springer Verlag. https://doi.org/10.1007/978-3-642-15775-2_10
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