Improved Bounds for the Crossing Numbers of Km, n and Kn

It has long been conjectured that the crossing number cr(Km,n) of the complete bipartite graph Km,n is equal to Z(m,n) := bm−1 2 cb2 cbn−1 2 cb2 c. Another long-standing conjecture is that the crossing number cr(Kn) of the complete graph Kn is equal to Z(n) := 4b2 cbn−1 2 cbn−2 2 cbn−3 2 c. In this talk, I will outline a new method that improves the asymptotic lower bounds to 0.83Z(m, n) and 0.83Z(n) respectively. This is follows from the improved lower bound cr(K7,n) ≥ 2.1796n2− 4.5n. The proof uses combinatorial ideas as well as quadratic optimization techniques. This is joint work with E. de Klerk, D.V. Pasechnik, R.B. Richter and G. Salazar.