260 research outputs found
Disjoint Empty Convex Pentagons in Planar Point Sets
In this paper we obtain the first non-trivial lower bound on the number of disjoint empty convex pentagons in planar points sets. We show that the number of disjoint empty convex pentagons in any set of n points in the plane, no three on a line, is at least ⌊5n/47⌋. This bound can be further improved to (3n−1)/28 for infinitely many n
Disjoint Empty Convex Pentagons in Planar Point Sets
Harborth [{\it Elemente der Mathematik}, Vol. 33 (5), 116--118, 1978] proved
that every set of 10 points in the plane, no three on a line, contains an empty
convex pentagon. From this it follows that the number of disjoint empty convex
pentagons in any set of points in the plane is least
. In this paper we prove that every set of 19
points in the plane, no three on a line, contains two disjoint empty convex
pentagons. We also show that any set of points in the plane, where
is a positive integer, can be subdivided into three disjoint convex regions,
two of which contains points each, and another contains a set of 9 points
containing an empty convex pentagon. Combining these two results, we obtain
non-trivial lower bounds on the number of disjoint empty convex pentagons in
planar points sets. We show that the number of disjoint empty convex pentagons
in any set of points in the plane, no three on a line, is at least
. This bound has been further improved to
for infinitely many .Comment: 23 pages, 28 figure
Happy endings for flip graphs
We show that the triangulations of a finite point set form a flip graph that
can be embedded isometrically into a hypercube, if and only if the point set
has no empty convex pentagon. Point sets of this type include convex subsets of
lattices, points on two lines, and several other infinite families. As a
consequence, flip distance in such point sets can be computed efficiently.Comment: 26 pages, 15 figures. Revised and expanded for journal publicatio
Every Large Point Set contains Many Collinear Points or an Empty Pentagon
We prove the following generalised empty pentagon theorem: for every integer
, every sufficiently large set of points in the plane contains
collinear points or an empty pentagon. As an application, we settle the
next open case of the "big line or big clique" conjecture of K\'ara, P\'or, and
Wood [\emph{Discrete Comput. Geom.} 34(3):497--506, 2005]
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 be a
set of points in the plane and in general position. A subset of ,
with four points, is called a -hole in if is in convex position and
its convex hull does not contain any point of in its interior. Two 4-holes
in are compatible if their interiors are disjoint. We show that
contains at least pairwise compatible 4-holes.
This improves the lower bound of which is implied by a
result of Sakai and Urrutia (2007).Comment: 17 page
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