Book drawings of complete bipartite graphs

4 We recall that a book with k pages consists of a straight line (the spine) and k half5 planes (the pages), such that the boundary of each page is the spine. If a graph is drawn 6 on a book with k pages in such a way that the vertices lie on the spine, and each edge 7 is contained in a page, the result is a k-page book drawing (or simply a k-page drawing). 8 The pagenumber of a graph G is the minimum k such that G admits a k-page embedding 9 (that is, a k-page drawing with no edge crossings). The k-page crossing number νk(G) 10 of G is the minimum number of crossings in a k-page drawing of G. We investigate the 11 pagenumbers and k-page crossing numbers of complete bipartite graphs. We find the 12 exact pagenumbers of several complete bipartite graphs, and use these pagenumbers 13 to find the exact k-page crossing number of Kk+1,n for k ∈ {3, 4, 5, 6}. We also prove 14 the general asymptotic estimate limk→∞ limn→∞ νk(Kk+1,n)/(2n /k) = 1. Finally, 15 we give general upper bounds for νk(Km,n), and relate these bounds to the k-planar 16 crossing numbers of Km,n and Kn. 17