Improved lower bounds on book crossing numbers of complete graphs
De Klerk, Etienne.
Pasechnik, Dmitrii V.
Date of Issue2013
School of Physical and Mathematical Sciences
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 νk(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 νk(Kn) for various values of k. We also use a maximum satisﬁability reformulation to obtain a computer-aided calculation of the exact value of νk(Kn) for several values of k and n. Finally, we investigate the best construction known for drawing Kn in k pages, calculate the resulting number of crossings, and discuss this upper bound in light of the new results reported in this paper.
SIAM journal on discrete mathematics
© 2013 Society for Industrial and Applied Mathematics. This paper was published in SIAM Journal on Discrete Mathematics and is made available as an electronic reprint (preprint) with permission of Society for Industrial and Applied Mathematics. The paper can be found at the following official DOI: [http://dx.doi.org/10.1137/120886777]. One print or electronic copy may be made for personal use only. Systematic or multiple reproduction, distribution to multiple locations via electronic or other means, duplication of any material in this paper for a fee or for commercial purposes, or modification of the content of the paper is prohibited and is subject to penalties under law.