New lower bounds on crossing numbers of $$K_{m,n}$$ from semidefinite programming

Abstract

AbstractIn this paper, we use semidefinite programming and representation theory to compute new lower bounds on the crossing number of the complete bipartite graph $$K_{m,n}$$ K m , 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 $$\mathop {\textrm{cr}}\limits (K_{10,n}) \ge 4.87057 n^2 - 10n$$ cr ( K 10 , n ) ≥ 4.87057 n 2 - 10 n , $$\mathop {\textrm{cr}}\limits (K_{11,n}) \ge 5.99939 n^2-12.5n$$ cr ( K 11 , n ) ≥ 5.99939 n 2 - 12.5 n , $$ \mathop {\textrm{cr}}\limits (K_{12,n}) \ge 7.25579 n^2 - 15n$$ cr ( K 12 , n ) ≥ 7.25579 n 2 - 15 n , $$\mathop {\textrm{cr}}\limits (K_{13,n}) \ge 8.65675 n^2-18n$$ cr ( K 13 , n ) ≥ 8.65675 n 2 - 18 n 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.