The 3-symmetric pseudolinear crossing number of K36

Abstract

Let n be a positive integer multiple of 3. A rectilinear drawing of the complete graph Kn in the plane is 3–symmetric if its underlying point set P is 3–symmetric, that is, if P is the disjoint union of three equal sized sets Q,ρ(Q) and ρ2(Q) such that ρ is a 2π/3 clockwise rotation around a suitable point in the plane. The 3–symmetric rectilinear crossing numbersym−cr3‾(Kn) of Kn is the minimum number of crossings in any 3–symmetric rectilinear drawing of Kn. In this paper, we extend these notions to the more general setting of pseudolinear drawings of Kn by defining the corresponding 3–symmetric pseudolinear crossing numbersym−cr3˜(Kn) of Kn, and show that sym−cr3˜(K36)=sym−cr3‾(K36)=21174.