Improved Bounds for the Crossing Numbers of Km,n and Kn

Abstract

It has been long-conjectured that the crossing number cr(K m,n ) of the complete bipartite graph K m,n equals the Zarankiewicz Number Z(m, n) :Another long-standing conjecture states that the crossing number cr(K n ) of the complete graph K n equals Z(n) := 1 4. In this paper we show the following improved bounds on the asymptotic ratios of these crossing numbers and their conjectured values:The previous best known lower bounds were 0.8m/(m − 1), 0.8, and 0.8, respectively. These improved bounds are obtained as a consequence of the new bound cr(K 7,n ) ≥ 2.1796n 2 − 4.5n. To obtain this improved lower bound for cr(K 7,n ), we use some elementary topological facts on drawings of K 2,7 to set up a quadratic program on 6! variables whose minimum p satisfies cr(K 7,n ) ≥ (p/2)n 2 − 4.5n, and then use state-ofthe-art quadratic optimization techniques combined with a bit of invariant theory of permutation groups to show that p ≥ 4.3593.

The complexity of optimizing over a simplex, hypercube or sphere: a short survey

“…For the simplex, and quadratic f , the applications include finding maximum stable sets in graphs, portfolio optimization, game theory, and population dynamics problems [see the review paper by Bomze (2002) and the references therein]. A recent application is the estimation of crossing numbers in certain classes of graphs (de Klerk et al 2006). Another application is testing matrix copositivity-recall that an n × n matrix A is called copositive if min x∈ n x T Ax = 0.…”

Copositive optimization – Recent developments and applications

“…In 1977, Pál Turán remembered [122] that he and Zarankiewicz worked on this problem in a concentration camp near Budapest, under forced labor, to minimize crossings of a railway system in brick production (every crossing involved danger of derailing, which was cruelly punished). Now [47] constructed a k × k matrix Q m (transposition distances of cyclic permutations) with k = (m − 1)! such that the crossing number can be estimated by the optimal value α Qm = min x ⊤ Q m x : x ∈ ∆ of the StQP based upon Q m : cr(K m,n ) ≥ n 2 n α Qm − (m − 1) 2 /4 .…”

“…Using the relation cr(K m,n ) ≥ m(m−1) r(r−1) cr(K r,n ) for all r ≤ m, and putting r = 9, these estimates have the asymptotic implication The best previously known constants were 0.8001 [104] and 0.83 [47]. We also obtain with Z(n) = 1 4 n 2 n−1 2 n−2 2 n−3 2 lim n→∞ cr(K n ) Z(n) ≥ lim n→∞ cr(K n,n ) Z(n, n) ≥ 0.8594 , and again the constant on the right-hand side is the best known up to now.…”

A new library of structured semidefinite programming instances

“…De Klerk et al [13] showed that one may obtain a lower bound on cr(K m,n ) via the optimal value of a suitable SDP problem.…”

“…[13] could solve the SDP problem for m = 7 by exploiting the algebraic structure of the matrix Q, to obtain the bound:…”