Crossing Numbers and Cutwidths

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. Wee assume that the drawing is good, i.e., incident edges do not cross, two edges cross at most once and at most two edges cross in a point of the plane. Leighton [13] proved that for any n-vertex graph G of bounded degree, its crossing number satisfies cr(G)+n =Ω (bw 2 (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 2 (G)) in [16, 20], 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 terms of its crossing number.