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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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
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