Improved lower bounds for the 2-page crossing numbers of K_{m,n} and K_n via semidefinite programming

TL;DR

This paper advances the understanding of 2-page crossing numbers for complete bipartite and complete graphs by proving asymptotic bounds and improving lower bounds using semidefinite programming techniques.

Contribution

It provides new asymptotic results for the 2-page crossing numbers of K_{m,n} and K_n, and improves the lower bounds for these crossing numbers.

Findings

Proved asymptotic equality for v_2(K_{m,n}) and Z(m,n) when m=7,8.

Established new lower bounds for v_2(K_n) with a ratio of at least 0.9253.

Extended known bounds for specific ranges of n, including n=15 to 18, 20, and 24.

Abstract

It has been long conjectured that the crossing numbers of the complete bipartite graph K_{m,n} and of the complete graph K_n equal Z(m,n) (the value conjectured by Zarankiewicz, who came up with a drawing reaching this value) and Z(n) :=Z(n,n-2)/4, respectively. In a 2-page drawing of a graph, the vertices are drawn on a straight line (the spine), and each edge is contained in one of the half-planes of the spine. The 2-page crossing number v_2(G) of a graph G is the minimum number of crossings in a 2-page drawing of G. Somewhat surprisingly, there are 2-page drawings of K_{m,n} (respectively, K_n) with exactly Z(m, n) (respectively, Z(n)) crossings, thus yielding the conjectures (I) v_2(Km,n) =Z(m,n), and (II) v_2(Kn) = Z(n). It is known that (I) holds for min{m, n} <=6, and that (II) holds for n<=14. In this paper we prove that (I) holds asymptotically (that is, lim_n v_2 (K_{m,n})/Z (m, n) = 1) for m=7 and 8. We also prove (II) for 15<=n<=18 and n=20,24, and establish the asymptotic estimate lim_n v_2(K_n)/Z(n) >= 0.9253. The previous best-known lower bound involved the constant 0.8594.