▁containing ▁at ▁most ▁two ▁vertices . ▁However , ▁like ▁the ▁standard ▁ 3 - edge - color ing ▁problem , ▁finding ▁a ▁color ing ▁of ▁this ▁type ▁is ▁N P - complete . ▁ ▁Total ▁color ing ▁is ▁a ▁form ▁of ▁color ing ▁that ▁comb ines ▁vertex ▁and ▁edge ▁color ing , ▁by ▁requiring ▁both ▁the ▁vertices ▁and ▁edges ▁to ▁be ▁colored . ▁Any ▁incident ▁pair ▁of ▁a ▁vertex ▁and ▁an ▁edge , ▁or ▁an ▁edge ▁and ▁an ▁edge , ▁must ▁have ▁distinct ▁colors , ▁as ▁must ▁any ▁two ▁adjacent ▁vertices . ▁It ▁has ▁been ▁conject ured ▁( com bin ing ▁V izing ' s ▁theorem ▁and ▁Bro oks ' ▁theorem ) ▁that ▁any ▁graph ▁has ▁a ▁total ▁color ing ▁in ▁which ▁the ▁number ▁of ▁colors ▁is ▁at ▁most ▁the ▁maximum ▁degree ▁plus ▁two , ▁but ▁this ▁remains ▁un pro ven . ▁ ▁If ▁a ▁ 3 - regular ▁graph ▁on ▁a ▁surface ▁is ▁ 3 - edge - color ed , ▁its ▁dual ▁graph ▁forms ▁a ▁tri ang ulation ▁of ▁the ▁surface ▁which ▁is ▁also ▁edge ▁colored ▁( although ▁not , ▁in ▁general , ▁properly ▁edge ▁colored ) ▁in ▁such ▁a ▁way ▁that ▁every ▁triangle ▁has ▁one ▁edge ▁of ▁each ▁color . ▁Other ▁color ings ▁and ▁orient ations ▁of ▁tri ang ulations , ▁with ▁other ▁local ▁constraints ▁on ▁how ▁the ▁colors ▁are ▁arranged ▁at ▁the ▁vertices ▁or ▁faces ▁of ▁the ▁tri ang ulation , ▁may ▁be ▁used ▁to ▁encode ▁several ▁types ▁of ▁geometric ▁object . ▁For ▁instance , ▁rect angular ▁sub div