ided ▁polygon . ▁For ▁each ▁color ▁class , ▁include ▁one ▁edge ▁from ▁the ▁center ▁to ▁one ▁of ▁the ▁polygon ▁vertices , ▁and ▁all ▁of ▁the ▁per pend icular ▁edges ▁connecting ▁pairs ▁of ▁polygon ▁vertices . ▁However , ▁when ▁ ▁is ▁odd , ▁ ▁colors ▁are ▁needed : ▁each ▁color ▁can ▁only ▁be ▁used ▁for ▁ ▁edges , ▁a ▁ ▁fraction ▁of ▁the ▁total . ▁ ▁Several ▁authors ▁have ▁studied ▁edge ▁color ings ▁of ▁the ▁odd ▁graphs , ▁- regular ▁graphs ▁in ▁which ▁the ▁vertices ▁represent ▁teams ▁of ▁ ▁players ▁selected ▁from ▁a ▁pool ▁of ▁ ▁players , ▁and ▁in ▁which ▁the ▁edges ▁represent ▁possible ▁pair ings ▁of ▁these ▁teams ▁( with ▁one ▁player ▁left ▁as ▁" odd ▁man ▁out " ▁to ▁ref eree ▁the ▁game ). ▁The ▁case ▁that ▁ ▁gives ▁the ▁well - known ▁Peters en ▁graph . ▁As ▁ ▁explains ▁the ▁problem ▁( for ▁), ▁the ▁players ▁wish ▁to ▁find ▁a ▁schedule ▁for ▁these ▁pair ings ▁such ▁that ▁each ▁team ▁plays ▁each ▁of ▁its ▁six ▁games ▁on ▁different ▁days ▁of ▁the ▁week , ▁with ▁Sund ays ▁off ▁for ▁all ▁teams ; ▁that ▁is , ▁formal izing ▁the ▁problem ▁mathemat ically , ▁they ▁wish ▁to ▁find ▁a ▁ 6 - edge - color ing ▁of ▁the ▁ 6 - regular ▁odd ▁graph ▁. ▁When ▁ ▁is ▁ 3 , ▁ 4 , ▁or ▁ 8 , ▁an ▁edge ▁color ing ▁of ▁ ▁requires ▁ ▁colors , ▁but ▁when ▁it ▁is ▁ 5 , ▁ 6 , ▁or ▁ 7 , ▁only ▁