Book drawings of complete bipartite graphs

Abstract

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 pagenumber 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 nu_k(G) of G is the minimum number of crossings in a k-page drawing of G. We investigate the pagenumbers and k-page crossing numbers of complete bipartite graphs. We find the exact pagenumbers of several complete bipartite graphs, and use these pagenumbers to find the exact k-page crossing number of K_{k+1,n} for 3<=k<=6. We also prove the general asymptotic estimate lim_{k->oo} lim_{n->oo} nu_k(K_{k+1,n})/(2n^2/k^2)=1. Finally, we give general upper bounds for nu_k(K_{m,n}), and relate these bounds to the k-planar crossing numbers of K_{m,n} and K_n.