An Improved Lower Bound for Crossing Numbers

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(bw(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 ∑ v∈G d 2 v = Ω(bw(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.