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Complete bipartite graphs

During World War II, Hungarian mathematician Pál Turán was forced to work in a brick factory, pushing wagon loads of bricks from kilns to storage sites. The factory had tracks from each kiln to each storage site, and the wagons were harder to push at the points where tracks crossed each other, from which Turán was led to ask his brick factory problem: how can the kilns, storage sites, and tracks be arranged to minimize the total number of crossings? Mathematically, the kilns and storage sites can be formalized as the vertices of a complete bipartite graph, with the tracks as its edges. A factory layout can be represented as a drawing of this graph, so the problem becomes:what is the minimum possible number of crossings in a drawing of a complete bipartite graph ?

Kazimierz Zarankiewicz attempted to solve Turán's brick factory problem; his proof contained an error, but he established a valid upper bound of

for the crossing number of the complete bipartite graph K. This bound has been conjectured to be the optimal number of crossings for all complete bipartite graphs.