The Maximum Number of Times the Same Distance Can Occur among the Vertices of a Convex n-gon Is O(nlogn)

Braβ, P.; Pach, János

doi:10.1006/jcta.2000.3133

Cited by 15 publications

(14 citation statements)

References 2 publications

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“…Our proof is based on the following lemma, which appeared in [4]. In that paper it was used to give a shorter proof of the theorem of Füredi [5], which states (using our notation) that max i n i = O(n log n).…”

Section: The Proofmentioning

confidence: 98%

On the number of occurrences of the kth smallest distance between points in convex position

Tomon

2015

Discrete Mathematics

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Section: The Proofmentioning

confidence: 98%

On the number of occurrences of the kth smallest distance between points in convex position

Tomon

2015

Discrete Mathematics

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“…Cn log n for a constant C, and is due to Füredi [10] and Braß and Pach [3]. The best lower bound, f (n) ≥ 2n−7, is due to Edelsbrunner and Hajnal [6].…”

Section: On the Chromatic Number Of Subsets Of Q × Rmentioning

confidence: 99%

On the Chromatic Number of Subsets of the Euclidean Plane

Axenovich

Choi

Lastrina

et al. 2012

Graphs and Combinatorics

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Abstract. The chromatic number of a subset of the real plane is the smallest number of colors assigned to the elements of that set such that no two points at distance 1 receive the same color. It is known that the chromatic number of the plane is between 4 and 7. In this note, we determine the bounds on the chromatic number for several classes of subsets of the plane such as extensions of the rational plane, sets in convex position, infinite strips, and parallel lines.

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“…The edge complexity of unit distance graphs on convex point sets have also been studied. The best lower bound is 2n − 7 [6] and the best upper bound is n log n [8,5]. It has been conjectured in [4] that the edge complexity of unit distance graphs on convex point sets is 2n.…”

Section: Introductionmentioning

confidence: 99%

On Locally Gabriel Geometric Graphs

Govindarajan

Khopkar

2014

Graphs and Combinatorics

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Let P be a set of n points in the plane. A geometric graph G on P is said to be locally Gabriel if for every edge (u, v) in G, the disk with u and v as diameter does not contain any points of P that are neighbors of u or v in G. A locally Gabriel graph is a generalization of Gabriel graph and is motivated by applications in wireless networks. Unlike a Gabriel graph, there is no unique locally Gabriel graph on a given point set since no edge in a locally Gabriel graph is necessarily included or excluded. Thus the edge set of the graph can be customized to optimize certain network parameters depending on the application. In this paper, we show the following combinatorial bounds on edge complexity and independent sets of locally Gabriel graphs:(i) For any n, there exists locally Gabriel graphs with Ω(n 5/4 ) edges. This improves upon the previous best bound of Ω(n 1+ 1 log log n ).(ii) For various subclasses of convex point sets, we show tight linear bounds on the maximum edge complexity of locally Gabriel graphs.(iii) For any locally Gabriel graph on any n point set, there exists an independent set of size Ω( √ n log n).

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The Maximum Number of Times the Same Distance Can Occur among the Vertices of a Convex n-gon Is O(nlogn)

Cited by 15 publications

References 2 publications

On the number of occurrences of the kth smallest distance between points in convex position

On the number of occurrences of the kth smallest distance between points in convex position

On the Chromatic Number of Subsets of the Euclidean Plane

On Locally Gabriel Geometric Graphs

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