Improved bounds for the crossing numbers of K_m,n and K_n

It has been long--conjectured that the crossing number cr(K_m,n) of the complete bipartite graph K_m,n equals the Zarankiewicz Number Z(m,n):= floor((m-1)/2) floor(m/2) floor((n-1)/2) floor(n/2). Another long--standing conjecture states that the crossing number cr(K_n) of the complete graph K_n equals Z(n):= floor(n/2) floor((n-1)/2) floor((n-2)/2) floor((n-3)/2)/4. In this paper we show the following improved bounds on the asymptotic ratios of these crossing numbers and their conjectured values: (i) for each fixed m >= 9, lim_{n->infty} cr(K_m,n)/Z(m,n) >= 0.83m/(m-1); (ii) lim_{n->infty} cr(K_n,n)/Z(n,n) >= 0.83; and (iii) lim_{n->infty} cr(K_n)/Z(n) >= 0.83. The previous best known lower bounds were 0.8m/(m-1), 0.8, and 0.8, respectively. These improved bounds are obtained as a consequence of the new bound cr(K_{7,n}) >= 2.1796n^2 - 4.5n. To obtain this improved lower bound for cr(K_{7,n}), we use some elementary topological facts on drawings of K_{2,7} to set up a quadratic program on 6! variables whose minimum p satisfies cr(K_{7,n}) >= (p/2)n^2 - 4.5n, and then use state--of--the--art quadratic optimization techniques combined with a bit of invariant theory of permutation groups to show that p >= 4.3593.Comment: LaTeX, 18 pages, 2 figure

On polynomials associated to Voronoi diagrams of point sets and crossing numbers

Three polynomials are defined for sets S of n points in general position in the plane: The Voronoi polynomial with coefficients the numbers of vertices of the order-k Voronoi diagrams of S, the circle polynomial with coefficients the numbers of circles through three points of S enclosing k points, and the E=k polynomial with coefficients the numbers of (at most k)-edges of S. We present several formulas for the rectilinear crossing number of S in terms of these polynomials and their roots. We also prove that the roots of the Voronoi polynomial lie on the unit circle if and only if S is in convex position. Further, we present bounds on the location of the roots of these polynomials.Postprint (published version

Geometric drawings of Kn with few crossings

AbstractWe give a new upper bound for the rectilinear crossing number cr¯(n) of the complete geometric graph Kn. We prove that cr¯(n)⩽0.380559(n4)+Θ(n3) by means of a new construction based on an iterative duplication strategy starting with a set having a certain structure of halving lines

New results on lower bounds for the number of k-facets

In this paper we present three different results dealing with the number of (≤ k)- facets of a set of points: (i) We give structural properties of sets in the plane that achieve the optimal lower bound 3_k+2 2 _ of (≤ k)-edges for a fixed k ≤ [n/3 ]− 1; (ii) We show that the new lower bound 3((k+2) 2 ) + 3((k−(n/ 3)+2) 2 ) for the number of (≤ k)-edges of a planar point set is optimal in the range [n/3] ≤ k ≤ [5n/12] − 1; (iii) We show that for k < n/4 the number of (≤ k)-facets of a set of n points in R3 in general position is at least 4((k+3 )3 ), and that this bound is tight in that range

Point sets that minimize (≤k)(\le k)-edges, 3-decomposable drawings, and the rectilinear crossing number of K30K_{30}

There are two properties shared by all known crossing-minimizing geometric drawings of $K_n$, for $n$ a multiple of 3. First, the underlying $n$-point set of these drawings has exactly $3\binom{k+2}{2}$ $(\le k)$-edges, for all $0\le k < n/3$. Second, all such drawings have the $n$ points divided into three groups of equal size; this last property is captured under the concept of 3-decomposability. In this paper we show that these properties are tightly related: every $n$-point set with exactly $3\binom{k+2}{2}$ $(\le k)$-edges for all $0\le k < n/3$, is 3-decomposable. As an application, we prove that the rectilinear crossing number of $K_{30}$ is 9726.Comment: 14 page

Compatible 4-Holes in Point Sets

Counting interior-disjoint empty convex polygons in a point set is a typical Erd\H{o}s-Szekeres-type problem. We study this problem for 4-gons. Let $P$ be a set of $n$ points in the plane and in general position. A subset $Q$ of $P$, with four points, is called a $4$-hole in $P$ if $Q$ is in convex position and its convex hull does not contain any point of $P$ in its interior. Two 4-holes in $P$ are compatible if their interiors are disjoint. We show that $P$ contains at least $\lfloor 5n/11\rfloor {-} 1$ pairwise compatible 4-holes. This improves the lower bound of $2\lfloor(n-2)/5\rfloor$ which is implied by a result of Sakai and Urrutia (2007).Comment: 17 page