Crossing number, pair-crossing number, and expansion

The crossing number crðGÞ of a graph G is the minimum possible number of edge crossings in a drawing of G in the plane, while the pair-crossing number pcrðGÞ is the smallest number of pairs of edges that cross in a drawing of G in the plane. While crðGÞXpcrðGÞ holds trivially, it is not known whether a strict inequality can ever occur (this question was raised by Mohar and Pach and Tóth). We aim at bounding crðGÞ in terms of pcrðGÞ:Using the methods of Leighton and Rao, Bhatt and Leighton, and Even, Guha and Schieber, we prove that crðGÞ 1⁄4 Oðlog nðpcrðGÞ þ ssqdðGÞÞÞ; where n 1⁄4 jVðGÞj and ssqdðGÞ 1⁄4 P vAVðGÞ degGðvÞ : One of the main steps is an analogy of the well-known lower bound crðGÞ 1⁄4 OðbðGÞÞ OðssqdðGÞÞ; where bðGÞ is the bisection width of G; that is, the smallest number of edges that have to be removed so that no component of the resulting graph has more than 2 3 n vertices. We show that pcrðGÞ 1⁄4 OðbðGÞ=log nÞ OðssqdðGÞÞ: We also prove by similar methods that a graph G with crossing number k 1⁄4 crðGÞ4C ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ssqdðGÞ p m log n has a nonplanar subgraph on at most O Dnm log 2 n k vertices, where m is the number of edges, D is the maximum degree in G; and C is a suitable sufficiently large constant. r 2004 Elsevier Inc. All rights reserved.