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Improved Lower Bounds for the 2-Page Crossing Numbers of $K_{m,n}$ and $K_n$ via Semidefinite Programming

2012
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SIAM Journal on Optimization
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It has been long conjectured that the crossing numbers of the complete bipartite graph K_{m,n} and of the complete graph K_n equal Z(m,n) (the value conjectured by Zarankiewicz, who came up with a drawing reaching this value) and Z(n) :=Z(n,n-2)/4, respectively. In a 2-page drawing of a graph, the vertices are drawn on a straight line (the spine), and each edge is contained in one of the half-planes of the spine. The 2-page crossing number v_2(G) of a graph G is the minimum number of crossings

doi:10.1137/110852206
fatcat:et7rh7htpjck5oycgmuauulzje