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On the crossing number of \(K_{m,n}\). (English) Zbl 1023.05039
Electron. J. Comb. 10, Research paper N8, 6 p. (2003); printed version J. Comb. 10, No. 3 (2003).
Summary: The best lower bound known on the crossing number of the complete bipartite graph is \(\text{cr}(K_{m,n}) \geq (1/5)m(m-1)\lfloor n/2\rfloor\lfloor(n-1)/2\rfloor.\)
We prove that \(\text{cr}(K_{m,n}) \geq (1/5)m(m-1)\lfloor n/2 \rfloor \lfloor (n-1)/2 \rfloor + 9.9 \times 10^{-6} m^2n^2\) for sufficiently large \(m\) and \(n\).

MSC:
05C10 Planar graphs; geometric and topological aspects of graph theory
05C35 Extremal problems in graph theory
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