The 3-symmetric pseudolinear crossing number of K 36

Let n be a positive integer multiple of 3. A rectilinear drawing of the complete graph K n 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 ρ 2 ( Q ) such that ρ is a 2 π / 3 clockwise rotation around a suitable point in the plane. The 3 –symmetric rectilinear crossing number sym − cr 3 ‾ ( K n ) of K n is the minimum number of crossings in any 3–symmetric rectilinear drawing of K n. In this paper, we extend these notions to the more general setting of pseudolinear drawings of K n by defining the corresponding 3 –symmetric pseudolinear crossing number sym − cr 3 ˜ ( K n ) of K n, and show that sym − cr 3 ˜ ( K 36 ) = sym − cr 3 ‾ ( K 36 ) = 21174.