The circular k-partite crossing number of Km, n

We define a new kind of crossing number which generalizes both the bipartite crossing number and the outerplanar crossing number. We calculate exact values of this crossing number for many complete bipartite graphs and also give a lower bound. 1 Preliminaries The bipartite crossing number of a bipartite graph G was defined by Watkins in [W] to be the minimum number of crossings over all bipartite drawings of G. A bipartite drawing of bipartite G is one in which the vertices of the parts V1 and V2 are placed respectively on two distinct parallel lines and then the edges of G are drawn as straight line segments joining appropriate pairs of vertices. The calculation and estimation of this number are of interest to those who study VLSI design, graph drawing algorithms, and/or topological graph theory. See [N] for a bibliography on the topic as well as some of the few known exact results. The outerplanar crossing number of a graph G, also known as the circular or convex crossing number of G, was defined by Kainen in [K] to be the minimum number of crossings taken over all plane drawings of G where the vertices lie on a circle and the edges are chords of that circle. The calculation and estimation of this number are of interest to the same audience as the bipartite crossing number. See [C] and [F] for an introduction and bibliography, and [F] and [R] for the few known exact results.