The Bipartite-Cylindrical Crossing Number of the Complete Bipartite Graph

Abstract

A bipartite-cylindrical drawing of the complete bipartite graph $$ K_{m,n} $$ is a drawing on the surface of a cylinder, where the vertices are placed on the boundaries of the cylinder, one vertex-partition per boundary, and the edges do not cross the boundaries. The bipartite-cylindrical crossing number of $$ K_{m,n}$$, denoted by $$ cr_\circledcirc (K_{m,n}) $$, is the minimum number of crossings among all bipartite-cylindrical drawings of $$ K_{m,n} $$. This problem is equivalent to minimizing the number of crossings in drawings of $$ K_{m,n} $$ in the plane where each of the vertex classes in the bipartition is placed on a circle and the edges do not cross the two circles. We determine $$ cr_\circledcirc (K_{m,n}) $$ and give explicit constructions that achieve this number. This in turn gives an alternative proof to the Harary–Hill Conjecture restricted to a subclass of cylindrical drawings of the complete graph.