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

  • Authors:

  • Éva Czabarka;Ondrej Sýkora;László A. Székely;Imrich Vrťo

  • Affiliations:

  • Department of Mathematics, University of South Carolina, Columbia, South Carolina 29208;(Deceased) Department of Computer Science, Loughborough University, Loughborough, Leicestershire LE11 3TU, UK;Department of Mathematics, University of South Carolina, Columbia, South Carolina 29208;Department of Informatics, Institute of Mathematics, Slovak Academy of Sciences, Dúbravská 9, 841 04 Bratislava, Slovak Republic

  • Venue:
  • Random Structures & Algorithms
  • Year:
  • 2008

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Abstract

The biplanar crossing number cr2(G) ofa graph G isminG1∪G2=G{cr(G1) + cr(G2)}, wherecr is the planar crossing number. We show thatcr2(G) ≤ (3-8)cr(G).Using this result recursively, we bound the thickness byΘ(G) - 2 ≤Kcr2(G)0.4057log2n with some constant K. A partitionrealizing this bound for the thickness can be obtained by apolynomial time randomized algorithm. We show that for any sizeexceeding a certain threshold, there exists a graph G ofthis size, which simultaneously has the following properties:cr(G) is roughly as large as it can be for any graphof that size, and cr2(G) is as small as itcan be for any graph of that size. The existence is shown using theprobabilistic method. © 2008 Wiley Periodicals, Inc. RandomStruct. Alg., 2008We dedicate this paper to our late colleague and friend, OndrejSýkora.