Adam Sheffer

Counting plane graphs: Perfect matchings, spanning cycles, and Kasteleyn's technique

We derive improved upper bounds on the number of crossing-free straight-edge spanning cycles (also known as Hamiltonian tours and simple polygonizations) that can be embedded over any specific set of N points in the plane. More specifically, we bound the ratio between the number of spanning cycles (or perfect matchings) that can be embedded over a point set and the number of triangulations that can be embedded over it. The respective bounds are O(1.8181^N) for cycles and O(1.1067^N) for matchings. These imply a new upper bound of O(54.543^N) on the number of crossing-free straight-edge spanning cycles that can be embedded over any specific set of N points in the plane (improving upon the previous best upper bound O(68.664^N)). Our analysis is based on a weighted variant of Kasteleyn@?s linear algebra technique.

Distinct distances on two lines

Let P_1 and P_2 be two sets of points in the plane, so that P_1 is contained in a line L_1, P_2 is contained in a line L_2, and L_1 and L_2 are neither parallel nor orthogonal. Then the number of distinct distances determined by the pairs of P_1xP_2 is \Omega(\min{|P_1|^{2/3}|P_2|^{2/3},|P_1|^2, |P_2|^2}). In particular, if |P_1|=|P_2|=m, then the number of these distinct distances is \Omega(m^{4/3}), improving upon the previous bound \Omega(m^{5/4}) of Elekes.

Counting plane graphs: flippability and its applications

We generalize the notions of flippable and simultaneously flippable edges in a triangulation of a set S of points in the plane into so called pseudo-simultaneously flippable edges. We prove a worst-case tight lower bound for the number of pseudosimultaneously flippable edges in a triangulation in terms of the number of vertices. We use this bound for deriving new upper bounds for the maximal number of crossing-free straight-edge graphs that can be embedded on any fixed set of N points in the plane. We obtain new upper bounds for the number of spanning trees and forests as well. Specifically, let tr(N) denote the maximum number of triangulations on a set of N points in the plane. Then we show (using the known bound tr(N) < 30N) that any N-element point set admits at most 6.9283N ċ tr(N) < 207.85N crossing-free straight-edge graphs, O(4.8795N) ċ tr(N) = O(146.39N) spanning trees, and O(5.4723N) ċ tr(N) = O(164.17N) forests. We also obtain upper bounds for the number of crossing-free straight-edge graphs that have fewer than cN or more than cN edges, for a constant parameter c, in terms of c and N.

Improved Bounds for Incidences Between Points and Circles

We establish an improved upper bound for the number of incidences between m points and n circles in three dimensions. The previous best known bound, originally established for the planar case and later extended to any dimension ≥ 2, is O *( m 2/3 n 2/3 + m 6/11 n 9/11 + m + n ), where the O *(⋅) notation hides polylogarithmic factors. Since all the points and circles may lie on a common plane (or sphere), it is impossible to improve the bound in ℝ 3 without first improving it in the plane. Nevertheless, we show that if the set of circles is required to be ‘truly three-dimensional’ in the sense that no sphere or plane contains more than q of the circles, for some q ≪ n , then for any ϵ > 0 the bound can be improved to \[ O\bigl(m^{3/7+\eps}n^{6/7} + m^{2/3+\eps}n^{1/2}q^{1/6} + m^{6/11+\eps}n^{15/22}q^{3/22} + m + n\bigr). \] For various ranges of parameters ( e.g. , when m = Θ( n ) and q = o ( n 7/9 )), this bound is smaller than the lower bound Ω*( m 2/3 n 2/3 + m + n ), which holds in two dimensions. We present several extensions and applications of the new bound. (i) For the special case where all the circles have the same radius, we obtain the improved bound O ( m 5/11+ϵ n 9/11 + m 2/3+ϵ n 1/2 q 1/6 + m + n ). (ii) We present an improved analysis that removes the subpolynomial factors from the bound when m = O ( n 3/2−ϵ ) for any fixed ϵ (iii) We use our results to obtain the improved bound O ( m 15/7 ) for the number of mutually similar triangles determined by any set of m points in ℝ 3 . Our result is obtained by applying the polynomial partitioning technique of Guth and Katz using a constant-degree partitioning polynomial (as was also recently used by Solymosi and Tao). We also rely on various additional tools from analytic, algebraic, and combinatorial geometry.

Bounds on the maximum multiplicity of some common geometric graphs

We obtain new lower and upper bounds for the maximum multiplicity of some weighted, and respectively non-weighted, common geometric graphs drawn on

Counting Plane Graphs: Cross-Graph Charging Schemes

n

Point-curve incidences in the complex plane

points in the plane in general position (with no three points collinear): perfect matchings, spanning trees, spanning cycles (tours), and triangulations. (i) We present a new lower bound construction for the maximum number of triangulations a set of

Few distinct distances implies no heavy lines or circles

n

Counting plane graphs: perfect matchings, spanning cycles, and Kasteleyn's technique

points in general position can have. In particular, we show that a generalized double chain formed by two almost convex chains admits Omega (8.65^n) different triangulations. This improves the bound Omega (8.48^n) achieved by the previous best construction, the double zig-zag chain studied by Aichholzer et al. (ii) We present a new lower bound of Omega(11.97^n) for the number of non-crossing spanning trees of the double chain composed of two convex chains. The previous bound, Omega(10.42^n), stood unchanged for more than 10 years. (iii) Using a recent upper bound of 30^n for the number of triangulations, due to Sharir and Sheffer, we show that n points in the plane in general position admit at most O(68.664^n) non-crossing spanning cycles. (iv) We derive exponential lower bounds for the number of maximum and minimum weighted geometric graphs (matchings, spanning trees, and tours). It was known that the number of longest non-crossing spanning trees of a point set can be exponentially large, and here we show that this can be also realized with points in convex position. For points in convex position we obtain tight bounds for the number of longest and shortest tours. We give a combinatorial characterization of the longest tours, which leads to an O(n log n) time algorithm for computing them.

Counting plane graphs: cross-graph charging schemes

We study cross-graph charging schemes for graphs drawn in the plane. These are charging schemes where charge is moved across vertices of different graphs. Such methods have been recently applied to obtain various properties of triangulations that are embedded over a fixed set of points in the plane. We show how this method can be generalized to obtain results for various other types of graphs that are embedded in the plane. Specifically, we obtain a new bound of