Improved Lower Bounds on Book Crossing Numbers of Complete 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 $k$-page crossing number $\nu_k(G)$ of a graph $G$ is the minimum number of crossings in a $k$-page drawing of $G$. In this paper we investigate the $k$-page crossing numbers of complete graphs. We use semidefinite programming techniques to give improved lower bounds on $\nu_k(K_n)$ for various values of $k$. We also use a maximum satisfiability reformulation to obtain a computer-aided calculation of the exact value of $\nu_k(K_n)$ for several values of $k$ and $n$. Finally, we investigate the best construction known for drawing $K_n$ in $k$ pages, calculate the resulting number of crossings, and discuss this upper bound in light of the new results reported in this paper.
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© 2013, Society for Industrial and Applied Mathematics.
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Submitted: 2 August 2012
Accepted: 15 January 2013
Published online: 4 April 2013
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