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 [] 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] ∑_{v ∈ G}d_v² = Ω(bw²(G)) in [,], where d_v 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.