Improved bounds for the crossing numbers of Km,n and Kn

• Main Authors: Klerk, Etienne de., Pasechnik, Dmitrii V., Maharry, J., Richter, R. B., Salazar, G.
• Other Authors: School of Physical and Mathematical Sciences
• Format: Article
• Language: English
• Published: 2011
• Subjects: DRNTU::Science::Mathematics::Applied mathematics
• Online Access: https://hdl.handle.net/10356/100823; http://hdl.handle.net/10220/6787
• Institution: Nanyang Technological University

It has been long conjectured that the crossing number cr(Km,n) of the complete bipartite graph Km,n equals the Zarankiewicz number Z(m, n) :=[(m-1)/2][m/2][(n-1)/2][n/2]. Another longstanding conjecture states that the crossing number cr(Kn) of the complete graph Kn equals Z(n):=(1/4)[n/2][(n-1)/2][(n-2)/2][(n-3)/2]. In this paper we show the following improved bounds on the asymptotic ratios of these crossing numbers and their conjectured values: (i) for each fixed m ≥ 9, limn→∞ cr(Km,n)/Z(m, n) ≥ 0.83m/(m − 1); (ii) limn→∞ cr(Kn,n)/Z(n, n) ≥ 0.83; and iii) limn→∞ cr(Kn)/Z(n) ≥ 0.83. 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(K7,n) ≥ 2.1796n2 −4.5n2. To obtain this improved lower bound for cr(K7,n), we use some elementary topological facts on drawings of K2,7 to set up a quadratic program on 6! variables whose minimum p satisfies cr(K7,n) ≥ (p/2)n2−4.5n, and then use state-of-the-art quadratic optimization techniques combined with a bit of invariant theory of permutation groups to show that p ≥ 4.3593.