Book drawings of complete bipartite graphs

TL;DR

This paper studies the pagenumbers and crossing numbers of complete bipartite graphs within the book embedding model, providing exact values for certain cases and asymptotic estimates for large graphs.

Contribution

It determines the exact pagenumbers of several complete bipartite graphs and derives the asymptotic behavior of their k-page crossing numbers.

Findings

Exact pagenumbers for specific complete bipartite graphs.

Exact k-page crossing numbers for K_{k+1,n} with 3<=k<=6.

Asymptotic estimate of crossing numbers as n and k grow large.

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.