Book drawings of complete bipartite graphs
de Klerk,E. Pasechnik,D.V. Salazar,Gelasio
de Klerk,E.
Pasechnik,D.V.
Salazar,Gelasio
Abstract
We recall that 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 page number 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 νk(G) of G is the minimum number of crossings in a k-page drawing of G. We investigate the page numbers andk-page crossing numbers of complete bipartite graphs. We find the exact page numbers of several complete bipartite graphs, and use these page numbers to find the exactk-page crossing number of Kk+1,n for k∈{3,4,5,6}. We also prove the general asymptotic estimate limk→∞limn→∞νk(Kk+1,n)/(2n^2/k^2)=1. Finally, we give general upper bounds for νk(Km,n), and relate these bounds to the k-planar crossing numbers of Km,n and Kn.
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2014-04-20
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de Klerk, E, Pasechnik, D V & Salazar, G 2014, 'Book drawings of complete bipartite graphs', Discrete Applied Mathematics, vol. 167, pp. 80-93. https://doi.org/10.1016/j.dam.2013.11.001
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info:eu-repo/semantics/restrictedAccess