A pattern of asymptotic vertex valency distributions in planar maps
Journal of Combinatorial Theory Series B
A better upper bound on the number of triangulations of a planar point set
Journal of Combinatorial Theory Series A
Studies in computational geometry motivated by mesh generation
Studies in computational geometry motivated by mesh generation
A lower bound on the number of triangulations of planar point sets
Computational Geometry: Theory and Applications
Journal of Combinatorial Theory Series B
Random triangulations of planar point sets
Proceedings of the twenty-second annual symposium on Computational geometry
Counting crossing-free structures
Proceedings of the twenty-eighth annual symposium on Computational geometry
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We study the expected number of interior vertices of degree i in a triangulation of a point set S, drawn uniformly at random from the set of all triangulations of S, and derive various bounds and inequalities for these expected values. One of our main results is: For any set S of N points in general position, and for any fixed i, the expected number of vertices of degree i in a random triangulation is at least γiN, for some fixed positive constant γi (assuming that N i and that at least some fixed fraction of the points are interior). We also present a new application for these expected values, using upper bounds on the expected number of interior vertices of degree 3 to get a new lower bound, Ω(2.4317N), for the minimal number of triangulations any N-element planar point set in general position must have. This improves the previously best known lower bound of Ω(2.33N).