  It has been long conjectured that the crossing numbers of the complete
bipartite graph K_{m,n} and of the complete graph K_n equal Z(m,n) (the value
conjectured by Zarankiewicz, who came up with a drawing reaching this value)
and Z(n) :=Z(n,n-2)/4, respectively.
  In a 2-page drawing of a graph, the vertices are drawn on a straight line
(the spine), and each edge is contained in one of the half-planes of the spine.
The 2-page crossing number v_2(G) of a graph G is the minimum number of
crossings in a 2-page drawing of G. Somewhat surprisingly, there are 2-page
drawings of K_{m,n} (respectively, K_n) with exactly Z(m, n) (respectively,
Z(n)) crossings, thus yielding the conjectures (I) v_2(Km,n) =Z(m,n), and (II)
v_2(Kn) = Z(n).
  It is known that (I) holds for min{m, n} <=6, and that (II) holds for n<=14.
In this paper we prove that (I) holds asymptotically (that is, lim_n v_2
(K_{m,n})/Z (m, n) = 1) for m=7 and 8.
  We also prove (II) for 15<=n<=18 and n=20,24, and establish the asymptotic
estimate lim_n v_2(K_n)/Z(n) >= 0.9253.
  The previous best-known lower bound involved the constant 0.8594.
