, ▁the ▁general ▁case ▁remains ▁open . ▁ ▁Lower ▁bounds ▁Since ▁ ▁and ▁ ▁are ▁is omorphic , ▁it ▁is ▁enough ▁to ▁consider ▁the ▁case ▁where ▁. ▁In ▁addition , ▁for ▁ ▁Zar ank iew icz ' s ▁construction ▁gives ▁no ▁cross ings , ▁which ▁of ▁course ▁cannot ▁be ▁best ed . ▁So ▁the ▁only ▁n ont rivial ▁cases ▁are ▁those ▁for ▁which ▁ ▁and ▁ ▁are ▁both ▁. ▁ ▁Zar ank iew icz ' s ▁attempted ▁proof ▁of ▁the ▁conject ure , ▁although ▁invalid ▁for ▁the ▁general ▁case ▁of ▁, ▁works ▁for ▁the ▁case ▁. ▁It ▁has ▁since ▁been ▁extended ▁to ▁other ▁small ▁values ▁of ▁, ▁and ▁the ▁Zar ank iew icz ▁conject ure ▁is ▁known ▁to ▁be ▁true ▁for ▁the ▁complete ▁b ipart ite ▁graphs ▁ ▁with ▁. ▁The ▁conject ure ▁is ▁also ▁known ▁to ▁be ▁true ▁for ▁, ▁, ▁and ▁. ▁If ▁a ▁counter example ▁exists , ▁that ▁is , ▁a ▁graph ▁ ▁requiring ▁fewer ▁cross ings ▁than ▁the ▁Zar ank iew icz ▁bound , ▁then ▁in ▁the ▁smallest ▁counter example ▁both ▁ ▁and ▁ ▁must ▁be ▁odd . ▁ ▁For ▁each ▁fixed ▁choice ▁of ▁, ▁the ▁truth ▁of ▁the ▁conject ure ▁for ▁all ▁ ▁can ▁be ▁verified ▁by ▁testing ▁only ▁a ▁finite ▁number ▁of ▁choices ▁of ▁. ▁More ▁generally , ▁it ▁has ▁been ▁proven ▁that ▁every ▁complete ▁b ipart ite ▁graph ▁requires ▁a ▁number ▁of ▁cross ings ▁that ▁is ▁( for ▁sufficiently ▁large ▁graphs ) ▁at ▁least ▁ 8 3 % ▁of ▁the ▁number ▁given ▁by ▁the ▁Zar ank