An Improved Lower Bound for Crossing Numbers

  • Hristo Djidjev
  • Imrich Vrt’o
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 2265)

Abstract

The crossing number of a graph G = (V,E), denoted by cr(G), is the smallest number of edge crossings in any drawing of G in the plane. Leighton [14] proved that for any n-vertex graph G of bounded degree, its crossing number satisfies cr(G) + n = Ω(bw2(G)), where bw(G) is the bisection width of G. The lower bound method was extended for graphs of arbitrary vertex degrees to cr\( (G) + \tfrac{1} {{16}}\sum _{\upsilon \in G} d_\upsilon ^2 = \Omega (bw^2 (G))\) in [15],[19], where d ν is the degree of any vertex ν. We improve this bound by showing that the bisection width can be replaced by a larger parameter — the cutwidth of the graph. Our result also yields an upper bound for the path-width of G in term of its crossing number.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2002

Authors and Affiliations

  • Hristo Djidjev
    • 1
  • Imrich Vrt’o
    • 2
  1. 1.Department of Computer ScienceWarwick UniversityCoventryUK
  2. 2.Department of InformaticsInstitute of MathematicsBratislavaSlovak Republic

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