Independence Numbers of Planar Contact Graphs

Swanepoel, De Wet

doi:10.1007/s00454-002-2897-y

Cited by 4 publications

(3 citation statements)

References 5 publications

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“…Pach and Tóth [148] obtained the upper bound α m (n, E 2 ) 5n/16 . Swanepoel [178] also showed the lower bound α m (n, X 2 ) n/(4 − ε), where ε > 0 depends on X 2 , for each X 2 with λ (X 2 ) 1. Most likely this assumption on X 2 is unnecessary.…”

Section: Chromatic Number and Independence Number Of Minimum-distancementioning

confidence: 92%

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Combinatorial Distance Geometry in Normed Spaces

Swanepoel

2018

Bolyai Society Mathematical Studies

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We survey problems and results from combinatorial geometry in normed spaces, concentrating on problems that involve distances. These include various properties of unit-distance graphs, minimum-distance graphs, diameter graphs, as well as minimum spanning trees and Steiner minimum trees. In particular, we discuss translative kissing (or Hadwiger) numbers, equilateral sets, and the Borsuk problem in normed spaces. We show how to use the angular measure of Peter Brass to prove various statements about Hadwiger and blocking numbers of convex bodies in the plane, including some new results. We also include some new results on thin cones and their application to distinct distances and other combinatorial problems for normed spaces.

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Section: Chromatic Number and Independence Number Of Minimum-distancementioning

confidence: 92%

“…This was observed by Pollack [151] for the Euclidean plane. Csizmadia [50] improved the Euclidean lower bound to α m (n, E 2 ) 9n/35 and Swanepoel [178] to α m (n, E 2 ) 8n/31. Pach and Tóth [148] obtained the upper bound α m (n, E 2 ) 5n/16 .…”

Section: Chromatic Number and Independence Number Of Minimum-distance...mentioning

confidence: 99%

Combinatorial Distance Geometry in Normed Spaces

Swanepoel

2018

Bolyai Society Mathematical Studies

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“…(The problem is still open. Swanepoel [12] improved Ricky's bound to 8n/31, while from the other direction the best known result can be found in [9].) Ricky often referred to this note as his "favorite paper".…”

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confidence: 99%

A Farewell to Ricky Pollack

Pach

2020

Discrete Comput Geom

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How are you, Ricky? Until the very end, whenever I called him, he answered on the most cheerful note: Wonderful! This was not simply a mannerism on his part. It was enough to meet Ricky once to realize that he was an incorrigible optimist. What was the source of his optimism? After almost 40 years, I have some clues, but no satisfactory answer. Ricky did not have an easy start as a mathematician. When his uncle found out about his academic ambitions, his first reaction was: Are you nuts? How much money does a professor make? I can hire half a dozen people at such salaries! Start a business! But Ricky had no interest in money. His parents and many of their friends were Communists. During the McCarthy era, they were harassed by the authorities because of their political views. Ricky was one of the "red diaper babies." He shared his parents' dreams and wanted to make a difference. He always liked puzzles and chess, he excelled in mathematics, and above all he loved people. To become a teacher or a professor seemed like a natural goal for him. He was accepted to the PhD program in mathematics at NYU and started to do research in number theory under the direction of Harold N. Shapiro. After defending his thesis, he joined NYU as a professor. It must have been difficult for him to stay so close to his thesis advisor, who was not only a brilliant mathematician and engaging educator, but also had an imposing personality. It was the golden era of analytic number theory, and the field was full of extraordinary talents. It required a die-hard optimist like Ricky not to be discouraged by the grandeur of their achievements and to continue to do research. After spending a sabbatical in Montreal with Willy Moser and reuniting with his intellectual twin brother, Eli Goodman, he switched direction: the two of them started a lifelong collaboration in combinatorial geometry. The rest is history. In an effort to understand and improve the Erdős-Szekeres theorem on convex polygons and to give a combinatorial description of other geometric configurations, they explored the notion of allowable sequences, which has become a János Pach

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Maximum Area Independent Sets in Disk Intersection Graphs

Bereg

Dumitrescu

Jiang

2010

Int. J. Comput. Geom. Appl.

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Maximum Independent Set (MIS) and its relative Maximum Weight Independent Set (MWIS) are well-known problems in combinatorial optimization; they are NP-hard even in the geometric setting of unit disk graphs. In this paper, we study the Maximum Area Independent Set (MAIS) problem, a natural restricted version of MWIS in disk intersection graphs where the weight equals the disk area. We obtain: (i) Quantitative bounds on the maximum total area of an independent set relative to the union area; (ii) Practical constant-ratio approximation algorithms for finding an independent set with a large total area relative to the union area.

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Independence Numbers of Planar Contact Graphs

Cited by 4 publications

References 5 publications

Combinatorial Distance Geometry in Normed Spaces

Combinatorial Distance Geometry in Normed Spaces

A Farewell to Ricky Pollack

Maximum Area Independent Sets in Disk Intersection Graphs

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