Biplanar crossing numbers. II. Comparing crossing numbers and biplanar crossing numbers using the probabilistic method

Abstract

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 cr 2 (G) ≤ (3/8) cr (G). Using this result recursively, we bound the thickness by Θ (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. © 2008 Wiley Periodicals, Inc.