Biplanar Crossing Numbers. Ii. Comparing Crossing Numbers And Biplanar Crossing Numbers Using The Probabilistic Method

The biplanar crossing number cr(2)(G) of a graph G is min(G1UG2=G){cr(G(1)) + cr(G(2))}, 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 Theta(G) - 2 <= Kcr(2)(G)0.4057 log(2)n 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 cr(2)(G) is as small as it can be for any graph of that size. The existence is shown using the probabilistic method. (C) 2008 Wiley Periodicals, Inc. Random Struct. Alg., 33, 480-496, 2008