▁both ▁of ▁them ▁showed , ▁it ▁is ▁always ▁possible ▁to ▁draw ▁the ▁complete ▁b ipart ite ▁graph ▁ ▁( a ▁graph ▁with ▁ ▁vertices ▁on ▁one ▁side , ▁ ▁vertices ▁on ▁the ▁other ▁side , ▁and ▁ ▁edges ▁connecting ▁the ▁two ▁sides ) ▁with ▁a ▁number ▁of ▁cross ings ▁equal ▁to ▁ ▁The ▁construction ▁is ▁simple : ▁place ▁ ▁vertices ▁on ▁the ▁- axis ▁of ▁the ▁plane , ▁avoid ing ▁the ▁origin , ▁with ▁equal ▁or ▁nearly - equal ▁numbers ▁of ▁points ▁to ▁the ▁left ▁and ▁right ▁of ▁the ▁- axis . ▁Similarly , ▁place ▁ ▁vertices ▁on ▁the ▁- axis ▁of ▁the ▁plane , ▁avoid ing ▁the ▁origin , ▁with ▁equal ▁or ▁nearly - equal ▁numbers ▁of ▁points ▁above ▁and ▁below ▁the ▁- axis . ▁Then , ▁connect ▁every ▁point ▁on ▁the ▁- axis ▁by ▁a ▁straight ▁line ▁segment ▁to ▁every ▁point ▁on ▁the ▁- axis . ▁ ▁However , ▁their ▁proofs ▁that ▁this ▁formula ▁is ▁optimal , ▁that ▁is , ▁that ▁there ▁can ▁be ▁no ▁draw ings ▁with ▁fewer ▁cross ings , ▁were ▁err one ous . ▁The ▁gap ▁was ▁not ▁discovered ▁until ▁eleven ▁years ▁after ▁publication , ▁nearly ▁simultaneously ▁by ▁Ger hard ▁Ring el ▁and ▁Paul ▁K ain en . ▁Nevertheless , ▁it ▁is ▁conject ured ▁that ▁Zar ank iew icz ' s ▁and ▁Urban ik ' s ▁formula ▁is ▁optimal . ▁This ▁has ▁come ▁to ▁be ▁known ▁as ▁the ▁Zar ank iew icz ▁crossing ▁number ▁conject ure . ▁Although ▁some ▁special ▁cases ▁of ▁it ▁are ▁known ▁to ▁be ▁true