New lower bounds on crossing numbers of K_m,n from permutation modules and semidefinite programming
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In 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, extending a method from de Klerk et al. [SIAM J. Discrete Math. 20 (2006), 189-202] and the subsequent reduction by De Klerk, Pasechnik and Schrijver [Math. Prog. Ser. A and B, 109 (2007) 613-624]. We exploit the full symmetry of the problem by developing a block-diagonalization of the underlying matrix algebra and use it to improve bounds on several concrete instances. Our results imply that cr(K_10,n) ≥ 4.87057 n^2 - 10n, cr(K_11,n) ≥ 5.99939 n^2-12.5n, cr(K_12,n) ≥ 7.25579 n^2 - 15n, cr(K_13,n) ≥ 8.65675 n^2-18n for all n. The latter three bounds are computed using a new relaxation of the original semidefinite programming bound, by only requiring one small matrix block to be positive semidefinite. Our lower bound on K_13,n implies that for each fixed m ≥ 13, lim_n →∞cr(K_m,n)/Z(m,n) ≥ 0.8878 m/(m-1). Here Z(m,n) is the Zarankiewicz number: the conjectured crossing number of K_m,n.
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