TY - JOUR

T1 - The 2-Page Crossing Number of Kn

AU - Ábrego, Bernardo M.

AU - Aichholzer, Oswin

AU - Fernández-Merchant, Silvia

AU - Ramos, Pedro

AU - Salazar, Gelasio

N1 - Funding Information:
O. Aichholzer is partially supported by the ESF EUROCORES programme EuroGIGA, CRP ComPoSe, under grant FWF [Austrian Fonds zur Förderung der Wissenschaftlichen Forschung] I648-N18. P. Ramos is partially supported by MEC grant MTM2011-22792 and by the ESF EUROCORES programme EuroGIGA, CRP ComPoSe, under grant EUI-EURC-2011-4306. G. Salazar is supported by CONACYT grant 106432. This work was initiated during the workshop Crossing Numbers Turn Useful, held at the Banff International Research Station (BIRS). The authors thank the BIRS authorities and staff for their support. The authors would also like to thank the referees for their invaluable comments and suggestions.

PY - 2013/6

Y1 - 2013/6

N2 - Around 1958, Hill described how to draw the complete graph Kn with, crossings, and conjectured that the crossing number cr(Kn) is exactly Z(n). This is also known as Guy's conjecture as he later popularized it. Towards the end of the century, substantially different drawings of Kn 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 Kn, denoted by ν2(Kn), is the minimum number of crossings determined by a 2-page book drawing of Kn. Since cr(Kn) ≤ ν2(Kn) and ν2(Kn) ≤ Z(n), a natural step towards Hill's Conjecture is the weaker conjecture ν2(Kn) = Z(n), popularized by Vrt'o. In this paper we develop a new technique to investigate crossings in drawings of Kn, and use it to prove that ν2(Kn) = 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 Kn. We also introduce the concept of ≤≤ k-edges as a useful generalization of ≤k-edges and extend a powerful theorem that expresses the number of crossings in a rectilinear drawing of Kn in terms of its number of ≤k-edges to the topological setting. Finally, we give a complete characterization of crossing minimal 2-page book drawings of Kn and show that, up to equivalence, they are unique for n even, but that there exist an exponential number of non homeomorphic such drawings for n odd.

AB - Around 1958, Hill described how to draw the complete graph Kn with, crossings, and conjectured that the crossing number cr(Kn) is exactly Z(n). This is also known as Guy's conjecture as he later popularized it. Towards the end of the century, substantially different drawings of Kn 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 Kn, denoted by ν2(Kn), is the minimum number of crossings determined by a 2-page book drawing of Kn. Since cr(Kn) ≤ ν2(Kn) and ν2(Kn) ≤ Z(n), a natural step towards Hill's Conjecture is the weaker conjecture ν2(Kn) = Z(n), popularized by Vrt'o. In this paper we develop a new technique to investigate crossings in drawings of Kn, and use it to prove that ν2(Kn) = 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 Kn. We also introduce the concept of ≤≤ k-edges as a useful generalization of ≤k-edges and extend a powerful theorem that expresses the number of crossings in a rectilinear drawing of Kn in terms of its number of ≤k-edges to the topological setting. Finally, we give a complete characterization of crossing minimal 2-page book drawings of Kn and show that, up to equivalence, they are unique for n even, but that there exist an exponential number of non homeomorphic such drawings for n odd.

KW - 2-Page drawing

KW - Book drawing

KW - Complete graph

KW - Crossing number

KW - Topological drawing

KW - Discrete and Computational Geometry

UR - http://www.scopus.com/inward/record.url?scp=84879285642&partnerID=8YFLogxK

U2 - 10.1007/s00454-013-9514-0

DO - 10.1007/s00454-013-9514-0

M3 - Article

AN - SCOPUS:84879285642

VL - 49

SP - 747

EP - 777

JO - Discrete & Computational Geometry

JF - Discrete & Computational Geometry

SN - 0179-5376

IS - 4

ER -