A Tight Bound on the Number of Geometric Permutations of Convex Fat Objects in R d

We show that the maximum number of geometric permutations of a set of n pairwise-disjoint convex and fat objects in R ^d is O(n ^d-1 ) . This generalizes the bound of Θ (n ^d-1 ) obtained by Smorodinsky et al. [5] on the number of geometric permutations of n pairwise-disjoint balls. Received August 22, 2000, and in revised form February 6, 2001. Online publication October 12, 2001.     

A Constant Bound for Geometric Permutations of Disjoint Unit Balls

Abstract. We prove that a set of n disjoint unit balls in R ^d admits at most four distinct geometric permutations, or line transversals, thus settling a long-standing conjecture in combinatorial geometry. The constant bound significantly improves upon the Θ (n ^d-1 ) bound for disjoint balls of unrestricted radii.     

A Constant Bound for Geometric Permutations of Disjoint Unit Balls

   Abstract. We prove that a set of n disjoint unit balls in R ^d admits at most four distinct geometric permutations, or line transversals, thus settling a long-standing conjecture in combinatorial geometry. The constant bound significantly improves upon the Θ (n ^d-1 ) bound for disjoint balls of unrestricted radii.     

Geometric Graphs with Few Disjoint Edges

A geometric graph is a graph drawn in the plane so that the vertices are represented by points in general position, the edges are represented by straight line segments connecting the corresponding points. Improving a result of Pach and T?rőcsik, we show that a geometric graph on n vertices with no k+1 pairwise disjoint edges has at most k ³ (n+1) edges. On the other hand, we construct geometric graphs with n vertices and approximately (3/2)(k-1)n edges, containing no k+1 pairwise disjoint edges. We also improve both the lower and upper bounds of Goddard, Katchalski, and Kleitman on the maximum number of edges in a geometric graph with no four pairwise disjoint edges. Received May 7, 1998, and in revised form March 24, 1999.     

Polytopes in Arrangements

Consider an arrangement of n hyperplanes in \real ^d . Families of convex polytopes whose boundaries are contained in the union of the hyperplanes are the subject of this paper. We aim to bound their maximum combinatorial complexity. Exact asymptotic bounds were known for the case where the polytopes are cells of the arrangement. Situations where the polytopes are pairwise openly disjoint have also been considered in the past. However, no nontrivial bound was known for the general case where the polytopes may have overlapping interiors, for d>2 . We analyze families of polytopes that do not share vertices. In \real ³ we show an O(k ^1/3 n ² ) bound on the number of faces of k such polytopes. We also discuss worst-case lower bounds and higher-dimensional versions of the problem. Among other results, we show that the maximum number of facets of k pairwise vertex-disjoint polytopes in \real ^d is Ω(k ^1/2 n ^d/2 ) which is a factor of away from the best known upper bound in the range n ^d-2 ≤ k ≤ n ^d . The case where 1≤ k ≤ n ^d-2 is completely resolved as a known Θ(kn) bound for cells applies here. Received September 20, 1999, and in revised form March 10, 2000. Online publication September 22, 2000.     

Ray Shooting Amidst Convex Polygons in 2D

We consider the problem of ray shooting in a two-dimensional scene consisting ofmconvex polygons with a total ofnedges. We present a data structure that requiresO(mn log m) space and preprocessing time and that answers a ray shooting query inO(log² m log² n) time. If the polygons are pairwise disjoint, the space and preprocessing time can be improved toO((m²+n)log m) andO((m²+n log n)log m), respectively. Our algorithm also works for a collection of disjoint simple polygons. We also show that if we allow onlyO(n) space, a ray shooting query among a collection of disjoint simple polygons can be answered in timeO(m/[formula]¹⁺ log² n) time, for any >0.     

Polynomial-time approximation schemes for packing and piercing fat objects

We consider two problems: given a collection of n fat objects in a fixed dimension, (1) ( packing) find the maximum subcollection of pairwise disjoint objects, and (2) ( piercing) find the minimum point set that intersects every object. Recently, Erlebach, Jansen, and Seidel gave a polynomial-time approximation scheme (PTAS) for the packing problem, based on a shifted hierarchical subdivision method. Using shifted quadtrees, we describe a similar algorithm for packing but with a smaller time bound. Erlebach et al.'s algorithm requires polynomial space. We describe a different algorithm, based on geometric separators, that requires only linear space. This algorithm can also be applied to piercing, yielding the first PTAS for that problem.     

The maximum number of ways to stabn convex nonintersecting sets in the plane is 2n−2

LetS be a collection ofn convex, closed, and pairwise nonintersecting sets in the Euclidean plane labeled from 1 ton. A pair of permutations $$\left\{ {(i_1 ,i_2 ,...,i_{n - 1} ,i_n ,),(i_n ,i_{n - 1} ,...,i_2 ,i_1 ,)} \right\}$$ is called ageometric permutation of S if there is a line that intersects all sets ofS in this order. We prove thatS can realize at most 2n?2 geometric permutations. This upper bound is tight.     

Almost Disjoint Triangles in 3-Space

Abstract. Two triangles are called almost disjoint if they are either disjoint or their intersection consists of one common vertex. Let f(n) denote the maximum number of pairwise almost disjoint triangles that can be found on some vertex set of n points in 3-space. Here we prove that f(n)=Ω(n ^3/2 ) .     

The different ways of stabbing disjoint convex sets

We construct a family ofn disjoint convex set in ^d having (n/(d–1)) ^d–1 geometric permutations. As well, we complete the enumeration problem for geometric permutations of families of disjoint translates of a convex set in the plane, settle the case for cubes in ^d , and construct a family ofd+1 translates in ^d admitting (d+1)!/2 geometric permutations.This research was partly supported by NSERC Grants A3062, A5137, and A8761.     

Almost Disjoint Triangles in 3-Space

   Abstract. Two triangles are called almost disjoint if they are either disjoint or their intersection consists of one common vertex. Let f(n) denote the maximum number of pairwise almost disjoint triangles that can be found on some vertex set of n points in 3-space. Here we prove that f(n)=Ω(n ^3/2 ) .     

Cutting Circles into Pseudo-Segments and Improved Bounds for Incidences% and Complexity of Many Faces

Abstract. We show that n arbitrary circles in the plane can be cut into O(n ^3/2+ɛ ) arcs, for any ɛ>0 , such that any pair of arcs intersects at most once. This improves a recent result of Tamaki and Tokuyama [20]. We use this result to obtain improved upper bounds on the number of incidences between m points and n circles. An improved incidence bound is also obtained for graphs of polynomials of any constant maximum degree.     

Cutting Circles into Pseudo-Segments and Improved Bounds for Incidences% and Complexity of Many Faces

   Abstract. We show that n arbitrary circles in the plane can be cut into O(n ^3/2+ɛ ) arcs, for any ɛ>0 , such that any pair of arcs intersects at most once. This improves a recent result of Tamaki and Tokuyama [20]. We use this result to obtain improved upper bounds on the number of incidences between m points and n circles. An improved incidence bound is also obtained for graphs of polynomials of any constant maximum degree.     

A note on a conjecture of Ryser

A special case of a conjecture of Ryser states that if a 3-partite 3-uniform hypergraph has at mostv pairwise disjoint edges then there is a set of vertices of cardinality at most 2v meeting all edges of the hypergraph. The best known upper bound for the size of such a set is (8/3)v, given by Tuza [7]. In this note we improve this to (5/2)v.     

The triples of geometric permutations for families of disjoint translates

A line meeting a family of pairwise disjoint convex sets induces two permutations of the sets. This pair of permutations is called a geometric permutation. We characterize the possible triples of geometric permutations for a family of disjoint translates in the plane. Together with earlier studies of geometric permutations this provides a complete characterization of realizable geometric permutations for disjoint translates.     

The Maximal Number of Geometric Permutations for n Disjoint Translates of a Convex Set in ℝ Is Ω(n)

A geometric permutation induced by a transversal line of a finite family of disjoint convex sets in ℝ^d is the order in which the transversal meets the members of the family. It is known that the maximal number of geometric permutations in families of n disjoint translates of a convex set in ℝ³ is 3. We prove that for d ≥ 3 the maximal number of geometric permutations for such families in ℝ^d is Ω(n).     

How to Cut Pseudoparabolas into Segments

Let Γ be a collection of unbounded x -monotone Jordan arcs intersecting at most twice each other, which we call pseudoparabolas, since two axis parallel parabolas intersect at most twice. We investigate how to cut pseudoparabolas into the minimum number of curve segments so that each pair of segments intersect at most once. We give an Ω( n ^4/3 ) lower bound and O(n ^5/3 ) upper bound on the number of cuts. We give the same bounds for an arrangement of circles. Applying the upper bound, we give an O(n ^23/12 ) bound on the complexity of a level in an arrangement of pseudoparabolas, and an O(n ^11/6 ) bound on the complexity of a combinatorially concave chain of pseudoparabolas. We also give some upper bounds on the number of transitions of the minimum weight matroid base when the weight of each element changes as a quadratic function of a single parameter. Received January 17, 1996, and in revised form November 7, 1996.     

Convexly independent sets

A family of pairwise disjoint compact convex sets is called convexly independent, if none of its members is contained in the convex hull of the union of the other members of the family. The main result of the paper gives an upper bound for the maximum cardinalityh(k, n) of a family of mutually disjoint compact convex sets such that any subfamily of at mostk members of is convexly independent, but no subfamily of sizen is.     

Inequalities for the first‐fit chromatic number

The First‐Fit (or Grundy) chromatic number of G, written as χ_FF(G), is defined as the maximum number of classes in an ordered partition of V(G) into independent sets so that each vertex has a neighbor in each set earlier than its own. The well‐known Nordhaus‐‐Gaddum inequality states that the sum of the ordinary chromatic numbers of an n‐vertex graph and its complement is at most n + 1. Zaker suggested finding the analogous inequality for the First‐Fit chromatic number. We show for n ≥ 10 that ?(5n + 2)/4? is an upper bound, and this is sharp. We extend the problem for multicolorings as well and prove asymptotic results for infinitely many cases. We also show that the smallest order of C₄‐free bipartite graphs with χ_FF(G) = k is asymptotically 2k² (the upper bound answers a problem of Zaker [9]). © 2008 Wiley Periodicals, Inc. J Graph Theory 59: 75–88, 2008     

Disjoint systems

A disjoint system of type (?, ?, k, n) is a collection ?? = {??₁,…, ??_m} of pairwise disjoint families of k-subsets of an n-element set satisfying the following condition. For every ordered pair ??_i and ??_j of distinct members of ?? and for every A ? ??_i there exists a B ? ??_j that does not intersect A. Let D_n (?, ?, k) denote the maximum possible cardinality of a disjoint system of type (?, ?, k, n). It is shown that for every fixed k ? 2,. This settles a problem of Ahlswede, Cai, and Zhang. Several related problems are considered as well.