Discrete & Computational Geometry

, Volume 49, Issue 4, pp 747–777 | Cite as

The 2-Page Crossing Number of \(K_{n}\)

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


Around 1958, Hill described how to draw the complete graph \(K_n\) with
$$\begin{aligned} Z(n) :=\frac{1}{4}\Big \lfloor \frac{n}{2}\Big \rfloor \Big \lfloor \frac{n-1}{2}\Big \rfloor \Big \lfloor \frac{n-2}{2}\Big \rfloor \Big \lfloor \frac{n-3}{2}\Big \rfloor \end{aligned}$$
crossings, and conjectured that the crossing number \({{\mathrm{cr}}}(K_{n})\) of \(K_n\) 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 \(K_{n}\) with \(Z(n)\) crossings were found. These drawings are 2-page book drawings, that is, drawings where all the vertices are on a line \(\ell \) (the spine) and each edge is fully contained in one of the two half-planes (pages) defined by \(\ell \). The 2-page crossing number of \(K_{n} \), denoted by \(\nu _{2}(K_{n})\), is the minimum number of crossings determined by a 2-page book drawing of \(K_{n}\). Since \({{\mathrm{cr}}}(K_{n}) \le \nu _{2}(K_{n})\) and \(\nu _{2}(K_{n}) \le Z(n)\), a natural step towards Hill’s Conjecture is the weaker conjecture \(\nu _{2}(K_{n}) = Z(n)\), popularized by Vrt’o. In this paper we develop a new technique to investigate crossings in drawings of \(K_{n}\), and use it to prove that \(\nu _{2}(K_{n}) = Z(n) \). 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 \({\le }{\le }k\)-edges as a useful generalization of \({\le }k\)-edges and extend a powerful theorem that expresses the number of crossings in a rectilinear drawing of \(K_{n}\) in terms of its number of \({\le }k\)-edges to the topological setting. Finally, we give a complete characterization of crossing minimal 2-page book drawings of \(K_{n}\) 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.


Crossing number Topological drawing Complete graph Book drawing 2-Page drawing 

Mathematics Subject Classification (2000)

05C10 68R10 52C10 57R15 


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Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  • Bernardo M. Ábrego
    • 1
  • Oswin Aichholzer
    • 2
  • Silvia Fernández-Merchant
    • 1
  • Pedro Ramos
    • 3
  • Gelasio Salazar
    • 4
  1. 1.Department of MathematicsCalifornia State UniversityNorthridgeUSA
  2. 2.Institute for Software TechnologyGraz University of TechnologyGrazAustria
  3. 3.Departamento de Física y MatemáticasUniversidad de AlcaláAlcalá de HenaresSpain
  4. 4.Instituto de FísicaUniversidad Autónoma de San Luis PotosíSan Luis PotosíMexico

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