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On Convex Geometric Graphs with no k+1 Pairwise Disjoint Edges
[article]

2015
*
arXiv
*
pre-print

A well-known result of Kupitz from 1982 asserts that the maximal number of

arXiv:1405.4019v2
fatcat:qjtg7vwa6jb7zfrzufn6uyn7ge
*edges*in a*convex**geometric**graph*(CGG)*on*n vertices that does not contain*k*+*1**pairwise**disjoint**edges*is kn (provided n>2k). ... We generalize our discussion to the following question: what is the maximal possible number f(n,*k*,q) of*edges*in a CGG*on*n vertices that does not contain*k*+*1**pairwise**disjoint**edges*, and, in addition, ... If the vertices are in*convex*position (i.e.,*no*vertex lies in the*convex*hull of the remaining vertices), the*graph*is called a*convex**geometric**graph*(CGG). ...##
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Ramsey-Type Results for Geometric Graphs, I

1997
*
Discrete & Computational Geometry
*

Under the same assumptions, we also prove that there exist (n +

doi:10.1007/pl00009317
fatcat:kopc2u4aczaqfaq3rjrjwd6rvq
*1*)/3*pairwise**disjoint*segments of the same color, and this bound is tight. ... Furthermore, improving an earlier result of Larman et al., we construct a family of m segments in the plane, which has*no*more than m log 4/ log 27 members that are either*pairwise**disjoint*or*pairwise*... Thus, there are n/2*pairwise**disjoint*red*edges**on*both sides of , contradicting our assumption that there are*no*n*disjoint**edges*of the same color in*K*3n−*1*. Case 2: n is odd. ...##
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Disjoint edges in geometric graphs

1989
*
Discrete & Computational Geometry
*

*disjoint*line segments. ... combinatorial geometry, we show that any configuration consisting of a set V of n points in general position in the plane and a set of 6n -5 closed straight line segments whose endpoints lie in V, contains three

*pairwise*... For

*k*-> 2, let f(

*k*n) denote the maximum number of

*edges*of a

*geometric*

*graph*

*on*n vertices that contains

*no*

*k*

*pairwise*

*disjoint*

*edges*. ...

##
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Thickness and Antithickness of Graphs
[article]

2018
*
arXiv
*
pre-print

This paper explores the relationship between the thickness and antithickness of a

arXiv:1708.04773v2
fatcat:7ktieooxb5glxkodmzmfgolqiu
*graph*, under various*graph*drawing models,*with*an emphasis*on*extremal questions. ... The "thickness" of a*graph*G is the minimum integer*k*such that in some drawing of G, the*edges*can be partitioned into*k*noncrossing subgraphs. ... Acknowledgement This research was initiated at the 2006 Bellairs Workshop*on*Computational Geometry organised by Godfried Toussaint. ...##
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Upper bounds on geometric permutations for convex sets

1990
*
Discrete & Computational Geometry
*

Let A be a family of n

doi:10.1007/bf02187777
fatcat:s7a3s6yrqvfc5etyoin57qvkxa
*pairwise**disjoint*compact*convex*sets in R a. Let =2Y,=o",_l(m-*1*)i . ... This bounds the number of*geometric*permutations*on*A by ½qbd ((~))ford>-3andby6nford=2. ... Given this triangulation G of A' we can embed the n*pairwise**disjoint**convex*polygons in A into n*pairwise**disjoint**convex*polygons*with*at most 12n*edges*. ...##
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On the chromatic number of some geometric type Kneser graphs

2005
*
Computational geometry
*

We estimate the chromatic number of

doi:10.1016/j.comgeo.2004.10.003
fatcat:ldlordd5irh4jclg62zmggje7a
*graphs*whose vertex set is the set of*edges*of a complete*geometric**graph**on*n points, and adjacency is defined in terms of*geometric**disjointness*or*geometric*intersection ... Acknowledgements We thank János Pach for pertinent remarks and suggestions, and an anonymous reviewer for simplifications made to*one*of our proofs. ... For any*k*< n/2, a*geometric**graphs**on*n vertices*with**no**k*+*1**pairwise**disjoint**edges*has at most 2 9*k*2 n*edges*. ...##
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On Geometric Graphs with No k Pairwise Parallel Edges

1998
*
Discrete & Computational Geometry
*

In this paper we show that, for any fixed

doi:10.1007/pl00009364
fatcat:fzrmoqehj5hlhkzqjzsipm24ny
*k*≥ 3, any*geometric**graph**on*n vertices*with**no**k**pairwise*parallel*edges*contains at most O(n)*edges*, and any*geometric**graph**on*n vertices*with**no**k**pairwise*... We also prove a conjecture by Kupitz that any*geometric**graph**on*n vertices*with**no*pair of parallel*edges*contains at most 2n − 2*edges*. ... In particular, their result implies that for any fixed*k*≥ 2 any*geometric**graph**on*n vertices*with**no**k**pairwise**disjoint**edges*contains at most O(n)*edges*. ...##
###
Some geometric applications of Dilworth's theorem

1994
*
Discrete & Computational Geometry
*

We settle an old question of Avital, Hanani, Erd~s, Kupitz, and Perles by showing that every

doi:10.1007/bf02574361
fatcat:syigyh5vujevfdv4proqmrp67i
*geometric**graph**with*n vertices and m > k4n*edges*contains*k*+*1**pairwise**disjoint**edges*. ... A*geometric**graph*is a*graph*drawn in the plane such that its*edges*are closed line segments and*no*three vertices are collinear. ... For every k, there exists C' k > 0 such that any*geometric**graph**with*n vertices and m >*k*4n*edges*has at least C'*k*m2k +*1*/n2k (*k*+*1*)-tuples of*pairwise**disjoint**edges*. ...##
###
The Beginnings of Geometric Graph Theory
[chapter]

2013
*
Bolyai Society Mathematical Studies
*

*Geometric*

*graphs*(topological

*graphs*) are

*graphs*drawn in the plane

*with*possibly crossing straight-line

*edges*(resp., curvilinear

*edges*). ... What is the maximum number of

*edges*that a

*geometric*or topological

*graph*of n vertices can have if it contains

*no*forbidden subconfiguration of a certain type? ... For

*convex*

*geometric*

*graphs*, that is, for

*geometric*

*graphs*whose vertices lie

*on*a closed

*convex*curve, Kupitz [Ku79] proved that this maximum is equal to (

*k*−

*1*)n, for all n > 2(

*k*−

*1*). ...

##
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On Geometric Graph Ramsey Numbers

2009
*
Graphs and Combinatorics
*

We also give a series of more general estimates

doi:10.1007/s00373-009-0847-7
fatcat:v2oahuv2qzhurgfsy7g7jo7cyy
*on*off-diagonal*geometric**graph*Ramsey numbers in the same spirit. ... Finally we investigate the existence of large noncrossing monochromatic matchings in multicoloured*geometric**graphs*. ... In view of Theorems*1*and 5, we only have to present a fourcolouring of the*edges*of a complete*convex**geometric**graph**on*6k − 5 vertices such that there are*no**k*(*geometrically*)*disjoint**edges*of the ...##
###
On grids in topological graphs

2014
*
Computational geometry
*

We also conjecture that every n-vertex

doi:10.1016/j.comgeo.2014.02.003
fatcat:go2xryyqejcp7l4cjp6sdi54rq
*geometric**graph**with**no*natural*k*-grid has O*k*(n)*edges*, but we can establish only an O*k*(n log 2 n) upper bound. ... It is known that every n-vertex topological*graph**with**no**k*-grid has O*k*(n)*edges*. ... We will only need to use the case p =*1*. The last tool we use is an upper bound*on*the number of*edges*of a*geometric**graph**with**no**k**pairwise**disjoint**edges*. ...##
###
On grids in topological graphs

2009
*
Proceedings of the 25th annual symposium on Computational geometry - SCG '09
*

We also conjecture that every n-vertex

doi:10.1145/1542362.1542430
dblp:conf/compgeom/AckermanFPS09
fatcat:lt6ikax4krb5laweowxm7ep4q4
*geometric**graph**with**no*natural*k*-grid has O*k*(n)*edges*, but we can establish only an O*k*(n log 2 n) upper bound. ... It is known that every n-vertex topological*graph**with**no**k*-grid has O*k*(n)*edges*. ... We will only need to use the case p =*1*. The last tool we use is an upper bound*on*the number of*edges*of a*geometric**graph**with**no**k**pairwise**disjoint**edges*. ...##
###
Edge-Removal and Non-Crossing Configurations in Geometric Graphs

2010
*
Discrete Mathematics & Theoretical Computer Science
*

We study the following extremal problem for

doi:10.46298/dmtcs.525
fatcat:2aevw6sqrbbm5bs6rxxmd5yveq
*geometric**graphs*: How many arbitrary*edges*can be removed from a complete*geometric**graph**with*n vertices such that the remaining*graph*still contains a certain ... In each case, we obtain tight bounds*on*the maximum number of removable*edges*. ... Equivalently, e*k*(n) is the maximum number of*edges*in a*geometric**graph**with*n vertices and*no**k*+*1**pairwise**disjoint**edges*. ...##
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A note on geometric 3-hypergraphs
[article]

2011
*
arXiv
*
pre-print

The two main theorems are

arXiv:1010.5716v3
fatcat:dic67fwnlbgh5kvyglgjz726rq
*1*) Every n-vertex*geometric*3-hypergraph in 2-space*with**no*three strongly crossing*edges*has at most O(n^2)*edges*, 2) Every n-vertex*geometric*3-hypergraph in 3-space*with**no*... two*disjoint**edges*has at most O(n^2)*edges*. ... Let ex d (D d*k*, n) denote the maximum*edges*that an n-vertex*geometric*d-hypergraph in d-space has*with**no**k**pairwise**disjoint**edges*. Then ex d (D d*k*, n) ≤ n d−(*1*/*k*) d−*1*. ...##
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Packing Plane Perfect Matchings into a Point Set
[article]

2015
*
arXiv
*
pre-print

What is the smallest number e

arXiv:1501.03686v1
fatcat:3znmvgkhhbafbko7ajzbkhpxt4
*k*(n) such that any*geometric**graph**with*n vertices and more than e*k*(n)*edges*contains*k*+*1**pairwise**disjoint**edges*, i.e., a plane matching of size at least*k*+*1*. ... Note that*k*≤ n/2 −*1*. By a result of Hopf and Pannwitz [23] and Erdős [15] , e*1*(n) = n, i.e., any*geometric**graph**with*n +*1**edges*contains a pair of*disjoint**edges*. ...
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