▁with ▁colours ▁ 1 . ▁. ▁. ▁t ▁is ▁an ▁interval ▁t ▁color ing ▁if ▁all ▁colours ▁are ▁used , ▁and ▁the ▁colours ▁of ▁edges ▁incident ▁to ▁each ▁vertex ▁of ▁G ▁are ▁distinct ▁and ▁form ▁an ▁interval ▁of ▁integers . ▁ ▁App lications ▁Edge ▁color ings ▁of ▁complete ▁graphs ▁may ▁be ▁used ▁to ▁schedule ▁a ▁round - ro bin ▁tournament ▁into ▁as ▁few ▁r ounds ▁as ▁possible ▁so ▁that ▁each ▁pair ▁of ▁compet itors ▁plays ▁each ▁other ▁in ▁one ▁of ▁the ▁r ounds ; ▁in ▁this ▁application , ▁the ▁vertices ▁of ▁the ▁graph ▁correspond ▁to ▁the ▁compet itors ▁in ▁the ▁tournament , ▁the ▁edges ▁correspond ▁to ▁games , ▁and ▁the ▁edge ▁colors ▁correspond ▁to ▁the ▁r ounds ▁in ▁which ▁the ▁games ▁are ▁played . ▁Similar ▁color ing ▁techniques ▁may ▁also ▁be ▁used ▁to ▁schedule ▁other ▁sports ▁pair ings ▁that ▁are ▁not ▁all - play - all ; ▁for ▁instance , ▁in ▁the ▁National ▁Football ▁League , ▁the ▁pairs ▁of ▁teams ▁that ▁will ▁play ▁each ▁other ▁in ▁a ▁given ▁year ▁are ▁determined , ▁based ▁on ▁the ▁teams ' ▁records ▁from ▁the ▁previous ▁year , ▁and ▁then ▁an ▁edge ▁color ing ▁algorithm ▁is ▁applied ▁to ▁the ▁graph ▁formed ▁by ▁the ▁set ▁of ▁pair ings ▁in ▁order ▁to ▁assign ▁games ▁to ▁the ▁week ends ▁on ▁which ▁they ▁are ▁played . ▁For ▁this ▁application , ▁V izing ' s ▁theorem ▁implies ▁that ▁no ▁matter ▁what ▁set ▁of ▁pair ings ▁is ▁chosen ▁( as ▁long ▁as ▁no ▁teams ▁play ▁each ▁other ▁twice ▁in ▁the ▁same ▁season ), ▁it