An Almost Optimal Bound on the Number of Intersections of Two Simple Polygons

Eyal Ackerman; Balázs Keszegh; Günter Rote

doi:10.1007/s00454-022-00438-0

An Almost Optimal Bound on the Number of Intersections of Two Simple Polygons

Eyal Ackerman, Balázs Keszegh, Günter Rote

Faculty of Natural Sciences

Research output: Contribution to journal › Article › peer-review

Abstract

What is the maximum number of intersections of the boundaries of a simple m-gon and a simple n-gon? This is a basic question in combinatorial geometry, and the answer is easy if at least one of m and n is even: If both m and n are even, then every pair of sides may cross and so the answer is mn. If exactly one polygon, say the n-gon, has an odd number of sides, it can intersect each side of the m-gon polygon at most n- 1 times; hence there are at most mn- m intersections. It is not hard to construct examples that meet these bounds. If both m and n are odd, the best known construction has mn- (m+ n) + 3 intersections, and it is conjectured that this is the maximum. However, the best known upper bound is only mn- (m+ ⌈ n/ 6 ⌉) , for m≥ n. We prove a new upper bound of mn- (m+ n) + C for some constant C, which is optimal apart from the value of C.

Original language	English
Pages (from-to)	1049-1077
Number of pages	29
Journal	Discrete and Computational Geometry
Volume	68
Issue number	4
DOIs	https://doi.org/10.1007/s00454-022-00438-0
State	Published - Dec 2022

Bibliographical note

Funding Information:
Eyal Ackerman: The main part of this work was performed during a visit to Freie Universität Berlin which was supported by the Freie Universität Alumni Program. Balázs Keszegh: Research supported by the Lendület program of the Hungarian Academy of Sciences (MTA), under the grant LP2017-19/2017, by the János Bolyai Research Scholarship of the Hungarian Academy of Sciences, by the National Research, Development and Innovation Office—NKFIH under the grant K 116769, K 132696, and FK 132060, and by the ÚNKP-21-5 New National Excellence Program of the Ministry for Innovation and Technology from the source of the National Research, Development and Innovation Fund.

Publisher Copyright:
© 2022, The Author(s).

Keywords

Combinatorial geometry
Ramsey theory
Simple polygon

ASJC Scopus subject areas

Theoretical Computer Science
Geometry and Topology
Discrete Mathematics and Combinatorics
Computational Theory and Mathematics

Access to Document

10.1007/s00454-022-00438-0

Cite this

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abstract = "What is the maximum number of intersections of the boundaries of a simple m-gon and a simple n-gon? This is a basic question in combinatorial geometry, and the answer is easy if at least one of m and n is even: If both m and n are even, then every pair of sides may cross and so the answer is mn. If exactly one polygon, say the n-gon, has an odd number of sides, it can intersect each side of the m-gon polygon at most n- 1 times; hence there are at most mn- m intersections. It is not hard to construct examples that meet these bounds. If both m and n are odd, the best known construction has mn- (m+ n) + 3 intersections, and it is conjectured that this is the maximum. However, the best known upper bound is only mn- (m+ ⌈ n/ 6 ⌉) , for m≥ n. We prove a new upper bound of mn- (m+ n) + C for some constant C, which is optimal apart from the value of C.",

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An Almost Optimal Bound on the Number of Intersections of Two Simple Polygons

Abstract

Bibliographical note

Keywords

ASJC Scopus subject areas

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