An Improved Lower Bound for Crossing Numbers

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.