The pagenumber of k-trees is O( k)

Abstract

A k-tree is a graph defined inductively in the following way: the complete graph K k is a k-tree, and if G is a k-tree, then the graph resulting from adding a new vertex adjacent to k vertices inducing a K k in G is also a k-tree. This paper examines the book-embedding problem for k-trees. A book embedding of a graph maps the vertices onto a line along the spine of the book and assigns the edges to pages of the book such that no two edges on the same page cross. The pagenumber of a graph is the minimum number of pages in a valid book embedding. In this paper, it is proven that the pagenumber of a k-tree is at most k+1. Furthermore, it is shown that there exist k-trees that require k pages. The upper bound leads to bounds on the pagenumber of a variety of classes of graphs for which no bounds were previously known.