New lower bounds on crossing numbers of K m,n from semidefinite programming

In this paper, we use semidefinite programming and representation theory to compute new lower bounds on the crossing number of the complete bipartite graph Km,n, extending a method from de Klerk et al. (SIAM J Discrete Math 20:189–202, 2006) and the subsequent reduction by De Klerk, Pasechnik and Schrijver (Math Prog Ser A and B 109:613–624, 2007). We exploit the full symmetry of the problem using a novel decomposition technique. This results in a full block-diagonalization of the underlying matrix algebra, which we use to improve bounds on several concrete instances. Our results imply that cr(K10,n) ≥ 4.87057n2 − 10n, cr(K11,n) ≥ 5.99939n2 − 12.5n, cr(K12,n) ≥ 7.25579n2 − 15n, cr(K13,n) ≥ 8.65675n2 − 18n for all n. The latter three bounds are computed using a new and well-performing relaxation of the original
semidefinite programming bound. This new relaxation is obtained by only requiring one small matrix block to be positive semidefinite.