Book drawings of complete bipartite graphs

We recall that a book with k pages consists of a straight line (the spine) and k half-planes (the pages), such that the boundary of each page is the spine. If a graph is drawn on a book with k pages in such a way that the vertices lie on the spine, and each edge is contained in a page, the result is a k-page book drawing (or simply a k-page drawing). The page number of a graph G is the minimum k such that G admits a k-page embedding (that is, a k-page drawing with no edge crossings). The k-page crossing number νk(G) of G is the minimum number of crossings in a k-page drawing of G. We investigate the page numbers andk-page crossing numbers of complete bipartite graphs. We find the exact page numbers of several complete bipartite graphs, and use these page numbers to find the exactk-page crossing number of Kk+1,n for k∈{3,4,5,6}. We also prove the general asymptotic estimate limk→∞limn→∞νk(Kk+1,n)/(2n^2/k^2)=1. Finally, we give general upper bounds for νk(Km,n), and relate these bounds to the k-planar crossing numbers of Km,n and Kn.