More on the crossing number of Kn: Monotone drawings

Bernardo M. Ábrego; Oswin Aichholzer; Silvia Fernández-Merchant; Pedro Ramos; Gelasio Salazar

doi:10.1016/j.endm.2013.10.064

More on the crossing number of K_n: Monotone drawings

Bernardo M. Ábrego^*, Oswin Aichholzer, Silvia Fernández-Merchant, Pedro Ramos, Gelasio Salazar

^*Corresponding author for this work

Graz University of Technology (90000)

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

Abstract

The Harary-Hill conjecture states that the minimum number of crossings in a drawing of the complete graph K_n is Z(n):=14⌊n2⌋⌊n-12⌋⌊n-22⌋⌊n-32⌋. This conjecture was recently proved for 2-page book drawings of K_n. As an extension of this technique, we prove the conjecture for monotone drawings of K_n, that is, drawings where all vertices have different x-coordinates and the edges are x-monotone curves.

Original language	English
Pages (from-to)	411-414
Number of pages	4
Journal	Electronic Notes in Discrete Mathematics
Volume	44
DOIs	https://doi.org/10.1016/j.endm.2013.10.064
Publication status	Published - 5 Nov 2013

Keywords

Complete graph
Crossing number
K-edge
Monotone drawing
Topological drawing

ASJC Scopus subject areas

Discrete Mathematics and Combinatorics
Applied Mathematics

Access to Document

10.1016/j.endm.2013.10.064

Cite this

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title = "More on the crossing number of Kn: Monotone drawings",

abstract = "The Harary-Hill conjecture states that the minimum number of crossings in a drawing of the complete graph Kn is Z(n):=14⌊n2⌋⌊n-12⌋⌊n-22⌋⌊n-32⌋. This conjecture was recently proved for 2-page book drawings of Kn. As an extension of this technique, we prove the conjecture for monotone drawings of Kn, that is, drawings where all vertices have different x-coordinates and the edges are x-monotone curves.",

keywords = "Complete graph, Crossing number, K-edge, Monotone drawing, Topological drawing, Discrete and Computational Geometry",

author = "{\'A}brego, {Bernardo M.} and Oswin Aichholzer and Silvia Fern{\'a}ndez-Merchant and Pedro Ramos and Gelasio Salazar",

note = "Funding Information: 1Email: bernardo.abrego@csun.edu 2 Partially supported by the ESF EUROCORES programme EuroGIGA, CRP ComPoSe, under grant FWF I648-N18. Email: oaich@ist.tugraz.at 3Email: silvia.fernandez@csun.edu 4 Partially supported by MEC grant MTM2011-22792 and ESF EUROCORES programme EuroGIGA, CRP ComPoSe, grant EUI-EURC-2011-4306. Email: pedro.ramos@uah.es 5 Supported by CONACYT grant 106432. Email: gsalazar@ifisica.uaslp.mx",

year = "2013",

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doi = "10.1016/j.endm.2013.10.064",

language = "English",

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journal = "Electronic Notes in Discrete Mathematics",

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T2 - Monotone drawings

AU - Ábrego, Bernardo M.

AU - Aichholzer, Oswin

AU - Fernández-Merchant, Silvia

AU - Ramos, Pedro

AU - Salazar, Gelasio

N1 - Funding Information: 1Email: bernardo.abrego@csun.edu 2 Partially supported by the ESF EUROCORES programme EuroGIGA, CRP ComPoSe, under grant FWF I648-N18. Email: oaich@ist.tugraz.at 3Email: silvia.fernandez@csun.edu 4 Partially supported by MEC grant MTM2011-22792 and ESF EUROCORES programme EuroGIGA, CRP ComPoSe, grant EUI-EURC-2011-4306. Email: pedro.ramos@uah.es 5 Supported by CONACYT grant 106432. Email: gsalazar@ifisica.uaslp.mx

PY - 2013/11/5

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N2 - The Harary-Hill conjecture states that the minimum number of crossings in a drawing of the complete graph Kn is Z(n):=14⌊n2⌋⌊n-12⌋⌊n-22⌋⌊n-32⌋. This conjecture was recently proved for 2-page book drawings of Kn. As an extension of this technique, we prove the conjecture for monotone drawings of Kn, that is, drawings where all vertices have different x-coordinates and the edges are x-monotone curves.

AB - The Harary-Hill conjecture states that the minimum number of crossings in a drawing of the complete graph Kn is Z(n):=14⌊n2⌋⌊n-12⌋⌊n-22⌋⌊n-32⌋. This conjecture was recently proved for 2-page book drawings of Kn. As an extension of this technique, we prove the conjecture for monotone drawings of Kn, that is, drawings where all vertices have different x-coordinates and the edges are x-monotone curves.

KW - Complete graph

KW - Crossing number

KW - K-edge

KW - Monotone drawing

KW - Topological drawing

KW - Discrete and Computational Geometry

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