On the length of longest alternating paths for multicoloured point sets in convex position

C. Merino; G. Salazar; J. Urrutia

doi:10.1016/j.disc.2006.03.035

Almost empty polygons

Studia Scientiarum Mathematicarum Hungarica ◽

10.1556/sscmath.40.2003.3.1 ◽

2003 ◽

Vol 40 (3) ◽

pp. 269-286 ◽

Cited By ~ 4

Author(s):

H. Nyklová

Keyword(s):

Exact Results ◽

Point Sets ◽

Planar Point ◽

Convex Position ◽

Vertex Set ◽

Point Set ◽

Empty Polygons

In this paper we study a problem related to the classical Erdos--Szekeres Theorem on finding points in convex position in planar point sets. We study for which n and k there exists a number h(n,k) such that in every planar point set X of size h(n,k) or larger, no three points on a line, we can find n points forming a vertex set of a convex n-gon with at most k points of X in its interior. Recall that h(n,0) does not exist for n = 7 by a result of Horton. In this paper we prove the following results. First, using Horton's construction with no empty 7-gon we obtain that h(n,k) does not exist for k = 2(n+6)/4-n-3. Then we give some exact results for convex hexagons: every point set containing a convex hexagon contains a convex hexagon with at most seven points inside it, and any such set of at least 19 points contains a convex hexagon with at most five points inside it.

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Drawing Hamiltonian Cycles with no Large Angles

The Electronic Journal of Combinatorics ◽

10.37236/2356 ◽

2012 ◽

Vol 19 (2) ◽

Cited By ~ 3

Author(s):

Adrian Dumitrescu ◽

János Pach ◽

Géza Tóth

Keyword(s):

Hamiltonian Cycle ◽

Hamiltonian Cycles ◽

Point Sets ◽

Convex Position ◽

Straight Line ◽

Open Region ◽

Point Set ◽

Rectifiable Curves ◽

Element Set

Let $n \geq 4$ be even. It is shown that every set $S$ of $n$ points in the plane can be connected by a (possibly self-intersecting) spanning tour (Hamiltonian cycle) consisting of $n$ straight-line edges such that the angle between any two consecutive edges is at most $2\pi/3$. For $n=4$ and $6$, this statement is tight. It is also shown that every even-element point set $S$ can be partitioned into at most two subsets, $S_1$ and $S_2$, each admitting a spanning tour with no angle larger than $\pi/2$. Fekete and Woeginger conjectured that for sufficiently large even $n$, every $n$-element set admits such a spanning tour. We confirm this conjecture for point sets in convex position. A much stronger result holds for large point sets randomly and uniformly selected from an open region bounded by finitely many rectifiable curves: for any $\epsilon>0$, these sets almost surely admit a spanning tour with no angle larger than $\epsilon$.

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A Bound on a Convexity Measure for Point Sets

International Journal of Computational Geometry & Applications ◽

10.1142/s0218195919500110 ◽

2019 ◽

Vol 29 (04) ◽

pp. 301-306

Author(s):

Danny Rorabaugh

Keyword(s):

Upper Bound ◽

Interior Angle ◽

Point Sets ◽

Planar Point ◽

Angle Measure ◽

Convex Position ◽

Point Set ◽

Set Of Points ◽

Convexity Measure

A planar point set is in convex position precisely when it has a convex polygonization, that is, a polygonization with maximum interior angle measure at most [Formula: see text]. We can thus talk about the convexity of a set of points in terms of its min-max interior angle measure. The main result presented here is a nontrivial upper bound of the min-max value in terms of the number of points in the set. Motivated by a particular construction, we also pose a natural conjecture for the best upper bound.

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More Turán-Type Theorems for Triangles in Convex Point Sets

The Electronic Journal of Combinatorics ◽

10.37236/7224 ◽

2019 ◽

Vol 26 (1) ◽

Author(s):

Boris Aronov ◽

Vida Dujmović ◽

Pat Morin ◽

Aurélien Ooms ◽

Luı́s Fernando Schultz Xavier da Silveira

Keyword(s):

Packing Problem ◽

Point Sets ◽

Convex Position ◽

Point Set ◽

Forbidden Configurations ◽

Convex Point

We study the following family of problems: Given a set of $n$ points in convex position, what is the maximum number triangles one can create having these points as vertices while avoiding certain sets of forbidden configurations. As forbidden configurations we consider all 8 ways in which a pair of triangles in such a point set can interact. This leads to 256 extremal Turán-type questions. We give nearly tight (within a $\log n$ factor) bounds for 248 of these questions and show that the remaining 8 questions are all asymptotically equivalent to Stein's longstanding tripod packing problem.

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NOTE ON THE NUMBER OF OBTUSE ANGLES IN POINT SETS

International Journal of Computational Geometry & Applications ◽

10.1142/s0218195914600012 ◽

2014 ◽

Vol 24 (03) ◽

pp. 177-181 ◽

Cited By ~ 1

Author(s):

RUY FABILA-MONROY ◽

CLEMENS HUEMER ◽

EULÀLIA TRAMUNS

Keyword(s):

Lower Bound ◽

Upper Bound ◽

General Position ◽

Crossing Number ◽

The Other ◽

Point Sets ◽

Convex Position ◽

Minimum Number

In 1979 Conway, Croft, Erdős and Guy proved that every set S of n points in general position in the plane determines at least [Formula: see text] obtuse angles and also presented a special set of n points to show the upper bound [Formula: see text] on the minimum number of obtuse angles among all sets S. We prove that every set S of n points in convex position determines at least [Formula: see text] obtuse angles, hence matching the upper bound (up to sub-cubic terms) in this case. Also on the other side, for point sets with low rectilinear crossing number, the lower bound on the minimum number of obtuse angles is improved.

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