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)+ 1/16 ΣvεG d2v = Ω (bw2 (G)) in [15,19], where dv is the degree of any vertex v. 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. © Springer-Verlag Berlin Heidelberg 2002.