research-article

The 2-page crossing number of K_n

Authors:

Bernardo Ábrego,

Oswin Aichholzer,

Silvia Fernández-Merchant,

Gelasio SalazarAuthors Info & Claims

SoCG '12: Proceedings of the twenty-eighth annual symposium on Computational geometry

Pages 397 - 404

https://doi.org/10.1145/2261250.2261310

Published: 17 June 2012 Publication History

Abstract

Around 1958, Hill conjectured that the crossing number CRg(K_n) of the complete graph KK_n is Z(n):=1/4 ⌊ n/2 ⌋ ⌊(n-1)/2⌋ ⌊ (n-2)/2 ⌋ ⌊ (n-3)/2 ⌋ and provided drawings of K_n with exactly Z(n) crossings. Towards the end of the century, substantially different drawings of K_n with Z(n) crossings were found. These drawings are 2-page book drawings, that is, drawings where all the vertices are on a line l (the spine) and each edge is fully contained in one of the two half-planes (pages) defined by l. The 2-page crossing number of K_n, denoted by ν₂(K_n), is the minimum number of crossings determined by a 2-page book drawing of K_n. Since CRG(K_n) ≤ ν₂(K_n) and ν₂(K_n) ≤ Z(n), a natural step towards Hill's Conjecture is the weaker conjecture ν₂(K_n) = Z(n), that was popularized by Vrt'o. In this paper we develop a novel and innovative technique to investigate crossings in drawings of K_n, and use it to prove that ν₂(K_n) = Z(n). To this end, we extend the inherent geometric definition of k-edges for finite sets of points in the plane to topological drawings of K_n. We also introduce the concept of ≤≤k-edges as a useful generalization of ≤k-edges. Finally, we extend a powerful theorem that expresses the number of crossings in a rectilinear drawing of K_n in terms of its number of k-edges to the topological setting.

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Cited By

Ábrego BFernández-Merchant SLagoda ERamos P(2019)On the Crossing Number of 2-Page Book Drawings of $$K_{n}$$ with Prescribed Number of Edges in Each PageGraphs and Combinatorics10.1007/s00373-019-02077-436:2(303-318)Online publication date: 4-Sep-2019
https://doi.org/10.1007/s00373-019-02077-4
Mutzel POettershagen L(2018)The Crossing Number of Seq-Shellable Drawings of Complete GraphsCombinatorial Algorithms10.1007/978-3-319-94667-2_23(273-284)Online publication date: 4-Jul-2018
https://doi.org/10.1007/978-3-319-94667-2_23
Ábrego BAichholzer OFernández-Merchant SRamos PSalazar G(2014)Shellable Drawings and the Cylindrical Crossing Number of $$K_{n}$$KnDiscrete & Computational Geometry10.1007/s00454-014-9635-052:4(743-753)Online publication date: 1-Dec-2014
https://dl.acm.org/doi/10.1007/s00454-014-9635-0
Show More Cited By

Index Terms

The 2-page crossing number of K_n
1. Mathematics of computing
  1. Discrete mathematics
    1. Graph theory

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cover image ACM Conferences

SoCG '12: Proceedings of the twenty-eighth annual symposium on Computational geometry

June 2012

436 pages

ISBN:9781450312998

DOI:10.1145/2261250

Program Chairs:
Tamal Dey
The Ohio State University, USA
,
Sue Whitesides
University of Victoria, Canada

Copyright © 2012 ACM.

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Published: 17 June 2012

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SoCG '12

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SoCG '12: Symposium on Computational Geometry 2012

June 17 - 20, 2012

North Carolina, Chapel Hill, USA

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Overall Acceptance Rate 625 of 1,685 submissions, 37%

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Cited By

Ábrego BFernández-Merchant SLagoda ERamos P(2019)On the Crossing Number of 2-Page Book Drawings of $$K_{n}$$ with Prescribed Number of Edges in Each PageGraphs and Combinatorics10.1007/s00373-019-02077-436:2(303-318)Online publication date: 4-Sep-2019
https://doi.org/10.1007/s00373-019-02077-4
Mutzel POettershagen L(2018)The Crossing Number of Seq-Shellable Drawings of Complete GraphsCombinatorial Algorithms10.1007/978-3-319-94667-2_23(273-284)Online publication date: 4-Jul-2018
https://doi.org/10.1007/978-3-319-94667-2_23
Ábrego BAichholzer OFernández-Merchant SRamos PSalazar G(2014)Shellable Drawings and the Cylindrical Crossing Number of $$K_{n}$$KnDiscrete & Computational Geometry10.1007/s00454-014-9635-052:4(743-753)Online publication date: 1-Dec-2014
https://dl.acm.org/doi/10.1007/s00454-014-9635-0
Ábrego BAichholzer OFernández-Merchant SRamos PSalazar G(2013)The 2-Page Crossing Number of $$K_{n}$$KnDiscrete & Computational Geometry10.1007/s00454-013-9514-049:4(747-777)Online publication date: 1-Jun-2013
https://dl.acm.org/doi/10.1007/s00454-013-9514-0
Faria Lde Figueiredo CRichter Rvrt’o I(2013)The Same Upper Bound for Both: The 2-Page and the Rectilinear Crossing Numbers of the n-CubeGraph-Theoretic Concepts in Computer Science10.1007/978-3-642-45043-3_22(249-260)Online publication date: 2013
https://doi.org/10.1007/978-3-642-45043-3_22

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