The Bipartite-Cylindrical Crossing Number of the Complete Bipartite Graph

A bipartite-cylindrical drawing of the complete bipartite graph Km,n\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ K_{m,n} $$\end{document} 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 Km,n\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ K_{m,n}$$\end{document}, denoted by cr⊚(Km,n)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ cr_\circledcirc (K_{m,n}) $$\end{document}, is the minimum number of crossings among all bipartite-cylindrical drawings of Km,n\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ K_{m,n} $$\end{document}. This problem is equivalent to minimizing the number of crossings in drawings of Km,n\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ K_{m,n} $$\end{document} 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⊚(Km,n)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ cr_\circledcirc (K_{m,n}) $$\end{document} 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.