TAILIEUCHUNG - Báo cáo toán học: "Counting Triangulations of Planar Point Sets"

Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí Department of Mathematic dành cho các bạn yêu thích môn toán học đề tài: Counting Triangulations of Planar Point Sets. | Counting Triangulations of Planar Point Sets Micha Sharir Blavatnik School of Computer Science Tel Aviv University Tel Aviv 69978 Israel and Courant Institute of Mathematical Sciences New York University New York NY 10012 USA michas@ Adam Sheffer Blavatnik School of Computer Science Tel Aviv University Tel Aviv 69978 Israel sheffera@ Submitted Mar 1 2010 Accepted Mar 24 2011 Published Mar 31 2011 Mathematics Subject Classifications 05C35 05C80 05C07 Abstract We study the maximal number of triangulations that a planar set of n points can have and show that it is at most 30n. This new bound is achieved by a careful optimization of the charging scheme of Sharir and Welzl 2006 which has led to the previous best upper bound of 43n for the problem. Moreover this new bound is useful for bounding the number of other types of planar . crossing-free straight-line graphs on a given point set. Specifically it can be used to derive new upper bounds for the number of planar graphs spanning cycles O spanning trees O and cycle-free graphs O . Keywords triangulations counting charging schemes crossing-free graphs. 1 Introduction A planar graph is a graph that can be drawn on the plane in such a way that its edges intersect only at their endpoints. A planar straight-line graph is an embedding of a planar Work on this paper was partially supported by Grants 155 05 and 338 09 from the Israel Science Fund. Work by Micha Sharir was also supported by NSF Grants CCF-05-14079 and CCF-08-30272 by Grant 2006 194 from the Binational Science Foundation and by the Hermann Minkowski-MINERVA Center for Geometry at Tel Aviv University. THE ELECTRONIC JOURNAL OF COMBINATORICS 18 2011 P70 1 graph in the plane such that its edges are mapped into straight line segments. In this paper we only consider planar straight-line graphs but refer to them as planar graphs for simplicity. Given a set S of points in the plane a triangulation of S is

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