2012/06/25 by Bernardo M. Abrego, Bernardo M. Ábrego, Oswin Aichholzer +9
Computer Science · Engineering · Mathematics · #Combinatorics (math.CO) #Computational Geometry (cs.CG) #Computational Geometry and Mesh Generation #Digital Image Processing Techniques #FOS: Computer and information sciences #FOS: Mathematics #cs.CG #graph theory and CDMA systems #math.CO
paper · pdf · doi:10.48550/arxiv.1206.5669
arxiv created 2012/06/25 · openalex publication_date 2012/06/25 · arxiv updated 2012/06/26 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Around 1958, Hill described how to draw the complete graph Kn with [Z(n) :=1/4\lfloor (n)/(2)\rfloor \lfloor (n-1)/(2)\rfloor \lfloor (n-2)/(2)% \rfloor \lfloor (n-3)/(2)\rfloor] crossings, and conjectured that the crossing number \crg (Kn) of 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 \crg(Kn) ≤ν2(Kn) and ν2(Kn) ≤ Z(n), a natural step towards Hill's Conjecture is the %(formally) weaker conjecture ν2(Kn) = Z(n), popularized by Vrt'o. %As far as we know, this natural %conjecture was first raised by Imrich Vrt'o in 2007. %Prior to this paper, results known for ν2(Kn) were basically %the same as for \crg (Kn). Here In this paper we develop a novel and innovative 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.