@inproceedings{e94511705f32453ab853bf0ad71c62bb,

title = "On the number of spanning trees a planar graph can have",

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).",

author = "K. Buchin and A. Schulz",

year = "2010",

doi = "10.1007/978-3-642-15775-2_10",

language = "English",

isbn = "978-3-642-15774-5",

series = "Lecture Notes in Computer Science",

publisher = "Springer",

pages = "110--121",

editor = "{Berg, de}, M. and U. Meyer",

booktitle = "Algorithms - ESA 2010 (18th Annual European Symposium, Liverpool, UK, September 6-8, 2010. Proceedings, Part I)",

address = "Germany",

}