Biplanar crossing numbers. II. Comparing crossing numbers and biplanar crossing numbers using the probabilistic method
Copyright policy )Biplanar crossing numbers. II. Comparing crossing numbers and biplanar crossing numbers using the probabilistic method
AbstractThe biplanar crossing number cr2(G) of a graph G is min{cr(G1) + cr(G2)}, where cr is the planar crossing number. We show that cr2(G) ≤ (3/8)cr(G). Using this result recursively, we bound the thickness by Θ(G) ‐ 2 ≤ Kcr2(G)0.4057 log2n with some constant K. A partition realizing this bound for the thickness can be obtained by a polynomial time randomized algorithm. We show that for any size exceeding a certain threshold, there exists a graph G of this size, which simultaneously has the following properties: cr(G) is roughly as large as it can be for any graph of that size, and cr2(G) is as small as it can be for any graph of that size. The existence is shown using the probabilistic method. © 2008 Wiley Periodicals, Inc. Random Struct. Alg., 2008
- University of South Carolina United States
- Loughborough University United Kingdom
- Slovak Academy of Sciences Slovakia
- University of South Carolina System United States
graph drawing, biplanar crossing number, Random graphs (graph-theoretic aspects), crossing number, thickness, Planar graphs; geometric and topological aspects of graph theory
graph drawing, biplanar crossing number, Random graphs (graph-theoretic aspects), crossing number, thickness, Planar graphs; geometric and topological aspects of graph theory
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