The 2-Page Crossing Number of Kn

Around 1958, Hill described how to draw the complete graph K n with Z ( n ) : = 1 4 ⌊ n 2 ⌋ ⌊ n − 1 2 ⌋ ⌊ n − 2 2 ⌋ ⌊ n − 3 2 ⌋ crossings, and conjectured that the crossing number 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 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 ν 2 ( K n ) , is the minimum number of crossings determined by a 2-page book drawing of K n . Since cr ( K n ) ≤ ν 2 ( K n ) and ν 2 ( K n ) ≤ Z ( n ) , a natural step towards Hill’s Conjecture is the weaker conjecture ν 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 ν 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 ≤ ≤ 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 K n 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 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.