Pagenumber of complete bipartite graphs

Abstract

AbstractGiven an ordering of the vertices of a graph around a circle, a page is a collection of edges forming noncrossing chords. A book embedding is a circular permutation of the vertices together with a partition of the edges into pages. The pagenumber t(G) (also called book thickness) is the minimum number of pages in a book embedding of G. We present a general construction showing t(Km,n) ⩽ ⌈(m + 2n)/4⌉, which we conjecture optimal. We prove a result suggesting this is optimal for m ⩾ 2n − 3. For the most difficult case m = n, we consider vertex permutations that are regular, i.e., place vertices from each partite set into runs of equal size. Book embeddings with such orderings require ⌈(7n − 2)/9⌉ pages, which is achievable. The general construction uses fewer pages, but with an irregular ordering.