The crossing number of a graph is the minimum number of edge crossings among all planar drawings of the graph. We will discuss the history of crossing number problems, including the current conjectures for the crossing number of the complete graph and the complete bipartite graph. We will also derive the cylindrical crossing number of the complete bipartite graph. This is the minimum number of crossings among all drawings of the complete bipartite graph where the two disjoint sets of vertices are placed on the top and bottom rims of a cylinder and every vertex at the top is connected to every vertex at the bottom by an edge drawn along the lateral surface of the cylinder.
Drawing and Counting with Athena - The Cylindrical Crossing Number of the Complete Bipartite Graph