▁a ▁drawing ▁of ▁a ▁complete ▁b ipart ite ▁graph . ▁The ▁problem ▁is ▁named ▁after ▁P ál ▁Tur án , ▁who ▁form ulated ▁it ▁while ▁being ▁forced ▁to ▁work ▁in ▁a ▁brick ▁factory ▁during ▁World ▁War ▁II . ▁ ▁A ▁drawing ▁method ▁found ▁by ▁Kaz im ierz ▁Zar ank iew icz ▁has ▁been ▁conject ured ▁to ▁give ▁the ▁correct ▁answer ▁for ▁every ▁complete ▁b ipart ite ▁graph , ▁and ▁the ▁statement ▁that ▁this ▁is ▁true ▁has ▁come ▁to ▁be ▁known ▁as ▁the ▁Zar ank iew icz ▁crossing ▁number ▁conject ure . ▁The ▁conject ure ▁remains ▁open , ▁with ▁only ▁some ▁special ▁cases ▁solved . ▁ ▁Origin ▁and ▁form ulation ▁During ▁World ▁War ▁II , ▁Hung arian ▁math ematic ian ▁P ál ▁Tur án ▁was ▁forced ▁to ▁work ▁in ▁a ▁brick ▁factory , ▁pushing ▁w agon ▁loads ▁of ▁br icks ▁from ▁kil ns ▁to ▁storage ▁sites . ▁The ▁factory ▁had ▁tracks ▁from ▁each ▁kil n ▁to ▁each ▁storage ▁site , ▁and ▁the ▁w ag ons ▁were ▁harder ▁to ▁push ▁at ▁the ▁points ▁where ▁tracks ▁crossed ▁each ▁other . ▁Tur án ▁was ▁inspired ▁by ▁this ▁situation ▁to ▁ask ▁how ▁the ▁factory ▁might ▁be ▁red es igned ▁to ▁minim ize ▁the ▁number ▁of ▁cross ings ▁between ▁these ▁tracks . ▁ ▁Mathemat ically , ▁this ▁problem ▁can ▁be ▁formal ized ▁as ▁asking ▁for ▁a ▁graph ▁drawing ▁of ▁a ▁complete ▁b ipart ite ▁graph , ▁whose ▁vertices ▁represent ▁kil ns ▁and ▁storage ▁sites , ▁and ▁whose ▁edges ▁represent ▁the ▁tracks ▁from ▁each ▁kil n ▁to ▁each ▁storage ▁site