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24 April 2024

» arxiv » 1607.0131

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Book crossing numbers of the complete graph and small local convex crossing numbers
Bernardo M. Ábrego ; Julia Dandurand ; Silvia Fernández-Merchant ; Evgeniya Lagoda ; Yakov Sapozhnikov ;
Date:	1 Jul 2016
Abstract:	A $ k $-page book drawing of a graph $ G $ is a drawing of $ G $ on $ k $ halfplanes with common boundary $ l $, a line, where the vertices are on $ l $ and the edges cannot cross $ l $. The $ k $-page book crossing number of the graph $ G $, denoted by $ u_k(G) $, is the minimum number of edge-crossings over all $ k $-page book drawings of $ G $. We improve the lower bounds on $ u_k(G) $ for all $ kgeq 15 $ and determine $ u_k(G) $ whenever $ 2 < n/k leq 3 $. Our proofs rely on bounding the number of edges in convex graphs with small local crossing numbers. In particular, we determine the maximum number of edges that a graph with local crossing number at most $ c $ can have for $ cleq 3 $.
Source:	arXiv, 1607.0131
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