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

Chaya Keller, Micha A. Perles
2015 arXiv   pre-print
A well-known result of Kupitz from 1982 asserts that the maximal number of 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).  ...

### Ramsey-Type Results for Geometric Graphs, I

G. Károlyi, J. Pach, G. Tóth
1997 Discrete & Computational Geometry
Under the same assumptions, we also prove that there exist (n + 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.  ...

### Disjoint edges in geometric graphs

N. Alon, P. Erdös
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.  ...

### Thickness and Antithickness of Graphs [article]

Vida Dujmović, David R. Wood
2018 arXiv   pre-print
This paper explores the relationship between the thickness and antithickness of a 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.  ...

### Upper bounds on geometric permutations for convex sets

Rephael Wenger
1990 Discrete & Computational Geometry
Let A be a family of n 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.  ...

### On the chromatic number of some geometric type Kneser graphs

G. Araujo, A. Dumitrescu, F. Hurtado, M. Noy, J. Urrutia
2005 Computational geometry
We estimate the chromatic number of 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.  ...

### On Geometric Graphs with No k Pairwise Parallel Edges

P. Valtr
1998 Discrete & Computational Geometry
In this paper we show that, for any fixed 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

J. Pach, J. Törőcsik
1994 Discrete & Computational Geometry
We settle an old question of Avital, Hanani, Erd~s, Kupitz, and Perles by showing that every 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]

János Pach
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 (k1)n, for all n > 2(k1).  ...

### On Geometric Graph Ramsey Numbers

Gyula Károlyi, Vera Rosta
2009 Graphs and Combinatorics
We also give a series of more general estimates 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

Eyal Ackerman, Jacob Fox, János Pach, Andrew Suk
2014 Computational geometry
We also conjecture that every n-vertex 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

Eyal Ackerman, Jacob Fox, János Pach, Andrew Suk
2009 Proceedings of the 25th annual symposium on Computational geometry - SCG '09
We also conjecture that every n-vertex 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

Oswin Aichholzer, Sergio Cabello, Ruy Fabila-Monroy, David Flores-Peñaloza, Thomas Hackl, Clemens Huemer, Ferran Hurtado, David R. Wood
2010 Discrete Mathematics & Theoretical Computer Science
We study the following extremal problem for 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.  ...

### A note on geometric 3-hypergraphs [article]

Andrew Suk
2011 arXiv   pre-print
The two main theorems are 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 .  ...

### Packing Plane Perfect Matchings into a Point Set [article]

Ahmad Biniaz, Prosenjit Bose, Anil Maheshwari, Michiel Smid
2015 arXiv   pre-print
What is the smallest number e 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  and Erdős  , e 1 (n) = n, i.e., any geometric graph with n + 1 edges contains a pair of disjoint edges.  ...
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