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❙❡❛r❝❤✐♥❣ ❢♦r ❛ ●r❛♣❤ ■s♦♠♦r♣❤✐s♠

❙❡❛r❝❤✐♥❣ ❢♦r ❛ ●r❛♣❤ ■s♦♠♦r♣❤✐s♠

• ✳❆✳◆♦s❦♦✈

✷✵✶✺

❙❡❛r❝❤✐♥❣ ❢♦r ❛ ●r❛♣❤ ■s♦♠♦r♣❤✐s♠

❖✉t❧✐♥❡

❋♦r♠✉❧❛t✐♦♥ ♦❢ t❤❡ ●■ ❙❡❛r❝❤✐♥❣ ❢♦r ❛ ●■ ❙●■ ✈❡rs✉s ●■ ❆❧❣♦r✐t❤♠ ❢♦r ❙●■ ❉✐s❡❛s❡ ❆ ❣✐❢t ❙♦❧✈✐♥❣ ●■ ❛✳❛✳s✉r❡❧② ❊✲t❡st ◗✉❡st✐♦♥s ❈♦♥❝❧✉s✐♦♥

❙❡❛r❝❤✐♥❣ ❢♦r ❛ ●r❛♣❤ ■s♦♠♦r♣❤✐s♠ ❋♦r♠✉❧❛t✐♦♥ ♦❢ t❤❡ ●■

❋♦r♠✉❧❛t✐♦♥ ♦❢ t❤❡ ●■

◮ ❆ ❣r❛♣❤ ✐s ❛ ♣❛✐r G = (V , E) ♦❢ s❡ts s✉❝❤ t❤❛t E ⊆

V 2

• ❀

t❤✉s✱ t❤❡ ❡❧❡♠❡♥ts ♦❢ E ✭t❤❡ ❡❞❣❡s✮ ❛r❡ ✷✲❡❧❡♠❡♥t s✉❜s❡ts ♦❢ V ✳ ❆♥ ✐s♦♠♦r♣❤✐s♠ ❢r♦♠ t♦ ✐s ❛ ❜✐❥❡❝t✐✈❡ ♠❛♣♣✐♥❣ ✭❛ ♣❡r♠✉t❛t✐♦♥✮ ❢r♦♠ t♦ s✉❝❤ t❤❛t t❤❡ ❡❞❣❡ ❝♦♥♥❡❝t✐♦♥s ❛r❡ r❡s♣❡❝t❡❞✱ ✐✳❡✳ ✳ ❛❧❧ ❣r❛♣❤s ♦♥ ✳ ▲❡t ❞❡♥♦t❡ t❤❡ s②♠♠❡tr✐❝ ❣r♦✉♣ ♦♥ ✳ ❛❝ts ❛❧s♦ ♦♥ ✲ t❤❡ s❡t ♦❢ ✷✲❡❧❡♠❡♥t s✉❜s❡ts ♦❢ ✳ ❋♦r ❡✈❡r② ♣❡r♠✉t❛t✐♦♥ ❛♥❞ ❡✈❡r② ❣r❛♣❤ ❞❡✜♥❡ ✳ ❚❤✉s ❛❝ts ♦♥ t❤❡ s❡t ❛♥❞ ✐s♦♠♦r♣❤✐s♠ ❝❧❛ss❡s ✐♥ ❛r❡ ♣r❡❝✐s❡❧② t❤❡ ♦r❜✐ts ♦❢ t❤✐s ❛❝t✐♦♥ ✳

❙❡❛r❝❤✐♥❣ ❢♦r ❛ ●r❛♣❤ ■s♦♠♦r♣❤✐s♠ ❋♦r♠✉❧❛t✐♦♥ ♦❢ t❤❡ ●■

❋♦r♠✉❧❛t✐♦♥ ♦❢ t❤❡ ●■

◮ ❆ ❣r❛♣❤ ✐s ❛ ♣❛✐r G = (V , E) ♦❢ s❡ts s✉❝❤ t❤❛t E ⊆

V 2

• ❀

t❤✉s✱ t❤❡ ❡❧❡♠❡♥ts ♦❢ E ✭t❤❡ ❡❞❣❡s✮ ❛r❡ ✷✲❡❧❡♠❡♥t s✉❜s❡ts ♦❢ V ✳

◮ ❆♥ ✐s♦♠♦r♣❤✐s♠ f ❢r♦♠ G t♦ H ✐s ❛ ❜✐❥❡❝t✐✈❡ ♠❛♣♣✐♥❣ ✭❛

♣❡r♠✉t❛t✐♦♥✮ ❢r♦♠ VG t♦ VH s✉❝❤ t❤❛t t❤❡ ❡❞❣❡ ❝♦♥♥❡❝t✐♦♥s ❛r❡ r❡s♣❡❝t❡❞✱ ✐✳❡✳ f (EG) = EH✳ ❛❧❧ ❣r❛♣❤s ♦♥ ✳ ▲❡t ❞❡♥♦t❡ t❤❡ s②♠♠❡tr✐❝ ❣r♦✉♣ ♦♥ ✳ ❛❝ts ❛❧s♦ ♦♥ ✲ t❤❡ s❡t ♦❢ ✷✲❡❧❡♠❡♥t s✉❜s❡ts ♦❢ ✳ ❋♦r ❡✈❡r② ♣❡r♠✉t❛t✐♦♥ ❛♥❞ ❡✈❡r② ❣r❛♣❤ ❞❡✜♥❡ ✳ ❚❤✉s ❛❝ts ♦♥ t❤❡ s❡t ❛♥❞ ✐s♦♠♦r♣❤✐s♠ ❝❧❛ss❡s ✐♥ ❛r❡ ♣r❡❝✐s❡❧② t❤❡ ♦r❜✐ts ♦❢ t❤✐s ❛❝t✐♦♥ ✳

❙❡❛r❝❤✐♥❣ ❢♦r ❛ ●r❛♣❤ ■s♦♠♦r♣❤✐s♠ ❋♦r♠✉❧❛t✐♦♥ ♦❢ t❤❡ ●■

❋♦r♠✉❧❛t✐♦♥ ♦❢ t❤❡ ●■

◮ ❆ ❣r❛♣❤ ✐s ❛ ♣❛✐r G = (V , E) ♦❢ s❡ts s✉❝❤ t❤❛t E ⊆

V 2

• ❀

t❤✉s✱ t❤❡ ❡❧❡♠❡♥ts ♦❢ E ✭t❤❡ ❡❞❣❡s✮ ❛r❡ ✷✲❡❧❡♠❡♥t s✉❜s❡ts ♦❢ V ✳

◮ ❆♥ ✐s♦♠♦r♣❤✐s♠ f ❢r♦♠ G t♦ H ✐s ❛ ❜✐❥❡❝t✐✈❡ ♠❛♣♣✐♥❣ ✭❛

♣❡r♠✉t❛t✐♦♥✮ ❢r♦♠ VG t♦ VH s✉❝❤ t❤❛t t❤❡ ❡❞❣❡ ❝♦♥♥❡❝t✐♦♥s ❛r❡ r❡s♣❡❝t❡❞✱ ✐✳❡✳ f (EG) = EH✳

◮ Gn = ❛❧❧ ❣r❛♣❤s ♦♥ [n] = {1, . . . , n}✳

▲❡t ❞❡♥♦t❡ t❤❡ s②♠♠❡tr✐❝ ❣r♦✉♣ ♦♥ ✳ ❛❝ts ❛❧s♦ ♦♥ ✲ t❤❡ s❡t ♦❢ ✷✲❡❧❡♠❡♥t s✉❜s❡ts ♦❢ ✳ ❋♦r ❡✈❡r② ♣❡r♠✉t❛t✐♦♥ ❛♥❞ ❡✈❡r② ❣r❛♣❤ ❞❡✜♥❡ ✳ ❚❤✉s ❛❝ts ♦♥ t❤❡ s❡t ❛♥❞ ✐s♦♠♦r♣❤✐s♠ ❝❧❛ss❡s ✐♥ ❛r❡ ♣r❡❝✐s❡❧② t❤❡ ♦r❜✐ts ♦❢ t❤✐s ❛❝t✐♦♥ ✳

❙❡❛r❝❤✐♥❣ ❢♦r ❛ ●r❛♣❤ ■s♦♠♦r♣❤✐s♠ ❋♦r♠✉❧❛t✐♦♥ ♦❢ t❤❡ ●■

❋♦r♠✉❧❛t✐♦♥ ♦❢ t❤❡ ●■

◮ ❆ ❣r❛♣❤ ✐s ❛ ♣❛✐r G = (V , E) ♦❢ s❡ts s✉❝❤ t❤❛t E ⊆

V 2

• ❀

t❤✉s✱ t❤❡ ❡❧❡♠❡♥ts ♦❢ E ✭t❤❡ ❡❞❣❡s✮ ❛r❡ ✷✲❡❧❡♠❡♥t s✉❜s❡ts ♦❢ V ✳

◮ ❆♥ ✐s♦♠♦r♣❤✐s♠ f ❢r♦♠ G t♦ H ✐s ❛ ❜✐❥❡❝t✐✈❡ ♠❛♣♣✐♥❣ ✭❛

♣❡r♠✉t❛t✐♦♥✮ ❢r♦♠ VG t♦ VH s✉❝❤ t❤❛t t❤❡ ❡❞❣❡ ❝♦♥♥❡❝t✐♦♥s ❛r❡ r❡s♣❡❝t❡❞✱ ✐✳❡✳ f (EG) = EH✳

◮ Gn = ❛❧❧ ❣r❛♣❤s ♦♥ [n] = {1, . . . , n}✳ ◮ ▲❡t Sn ❞❡♥♦t❡ t❤❡ s②♠♠❡tr✐❝ ❣r♦✉♣ ♦♥ [n]✳ Sn ❛❝ts ❛❧s♦ ♦♥ 2V

✲ t❤❡ s❡t ♦❢ ✷✲❡❧❡♠❡♥t s✉❜s❡ts ♦❢ V ✳ ❋♦r ❡✈❡r② ♣❡r♠✉t❛t✐♦♥ f ∈ Sn ❛♥❞ ❡✈❡r② ❣r❛♣❤ G = ([n], E) ❞❡✜♥❡ fG = ([n], fE)✳ ❚❤✉s ❛❝ts ♦♥ t❤❡ s❡t ❛♥❞ ✐s♦♠♦r♣❤✐s♠ ❝❧❛ss❡s ✐♥ ❛r❡ ♣r❡❝✐s❡❧② t❤❡ ♦r❜✐ts ♦❢ t❤✐s ❛❝t✐♦♥ ✳

❙❡❛r❝❤✐♥❣ ❢♦r ❛ ●r❛♣❤ ■s♦♠♦r♣❤✐s♠ ❋♦r♠✉❧❛t✐♦♥ ♦❢ t❤❡ ●■

❋♦r♠✉❧❛t✐♦♥ ♦❢ t❤❡ ●■

◮ ❆ ❣r❛♣❤ ✐s ❛ ♣❛✐r G = (V , E) ♦❢ s❡ts s✉❝❤ t❤❛t E ⊆

V 2

• ❀

t❤✉s✱ t❤❡ ❡❧❡♠❡♥ts ♦❢ E ✭t❤❡ ❡❞❣❡s✮ ❛r❡ ✷✲❡❧❡♠❡♥t s✉❜s❡ts ♦❢ V ✳

◮ ❆♥ ✐s♦♠♦r♣❤✐s♠ f ❢r♦♠ G t♦ H ✐s ❛ ❜✐❥❡❝t✐✈❡ ♠❛♣♣✐♥❣ ✭❛

♣❡r♠✉t❛t✐♦♥✮ ❢r♦♠ VG t♦ VH s✉❝❤ t❤❛t t❤❡ ❡❞❣❡ ❝♦♥♥❡❝t✐♦♥s ❛r❡ r❡s♣❡❝t❡❞✱ ✐✳❡✳ f (EG) = EH✳

◮ Gn = ❛❧❧ ❣r❛♣❤s ♦♥ [n] = {1, . . . , n}✳ ◮ ▲❡t Sn ❞❡♥♦t❡ t❤❡ s②♠♠❡tr✐❝ ❣r♦✉♣ ♦♥ [n]✳ Sn ❛❝ts ❛❧s♦ ♦♥ 2V

✲ t❤❡ s❡t ♦❢ ✷✲❡❧❡♠❡♥t s✉❜s❡ts ♦❢ V ✳ ❋♦r ❡✈❡r② ♣❡r♠✉t❛t✐♦♥ f ∈ Sn ❛♥❞ ❡✈❡r② ❣r❛♣❤ G = ([n], E) ❞❡✜♥❡ fG = ([n], fE)✳

◮ ❚❤✉s Sn ❛❝ts ♦♥ t❤❡ s❡t Gn ❛♥❞ ✐s♦♠♦r♣❤✐s♠ ❝❧❛ss❡s ✐♥ Gn ❛r❡

♣r❡❝✐s❡❧② t❤❡ ♦r❜✐ts ♦❢ t❤✐s ❛❝t✐♦♥ Sn Gn✳

❙❡❛r❝❤✐♥❣ ❢♦r ❛ ●r❛♣❤ ■s♦♠♦r♣❤✐s♠ ❋♦r♠✉❧❛t✐♦♥ ♦❢ t❤❡ ●■

❋♦r♠✉❧❛t✐♦♥ ♦❢ t❤❡ ●■

◮ ❆ ❣r❛♣❤ ✐s ❛ ♣❛✐r G = (V , E) ♦❢ s❡ts s✉❝❤ t❤❛t E ⊆

V 2

• ❀

t❤✉s✱ t❤❡ ❡❧❡♠❡♥ts ♦❢ E ✭t❤❡ ❡❞❣❡s✮ ❛r❡ ✷✲❡❧❡♠❡♥t s✉❜s❡ts ♦❢ V ✳

◮ ❆♥ ✐s♦♠♦r♣❤✐s♠ f ❢r♦♠ G t♦ H ✐s ❛ ❜✐❥❡❝t✐✈❡ ♠❛♣♣✐♥❣ ✭❛

♣❡r♠✉t❛t✐♦♥✮ ❢r♦♠ VG t♦ VH s✉❝❤ t❤❛t t❤❡ ❡❞❣❡ ❝♦♥♥❡❝t✐♦♥s ❛r❡ r❡s♣❡❝t❡❞✱ ✐✳❡✳ f (EG) = EH✳

◮ Gn = ❛❧❧ ❣r❛♣❤s ♦♥ [n] = {1, . . . , n}✳ ◮ ▲❡t Sn ❞❡♥♦t❡ t❤❡ s②♠♠❡tr✐❝ ❣r♦✉♣ ♦♥ [n]✳ Sn ❛❝ts ❛❧s♦ ♦♥ 2V

✲ t❤❡ s❡t ♦❢ ✷✲❡❧❡♠❡♥t s✉❜s❡ts ♦❢ V ✳ ❋♦r ❡✈❡r② ♣❡r♠✉t❛t✐♦♥ f ∈ Sn ❛♥❞ ❡✈❡r② ❣r❛♣❤ G = ([n], E) ❞❡✜♥❡ fG = ([n], fE)✳

◮ ❚❤✉s Sn ❛❝ts ♦♥ t❤❡ s❡t Gn ❛♥❞ ✐s♦♠♦r♣❤✐s♠ ❝❧❛ss❡s ✐♥ Gn ❛r❡

♣r❡❝✐s❡❧② t❤❡ ♦r❜✐ts ♦❢ t❤✐s ❛❝t✐♦♥ Sn Gn✳

❙❡❛r❝❤✐♥❣ ❢♦r ❛ ●r❛♣❤ ■s♦♠♦r♣❤✐s♠ ❋♦r♠✉❧❛t✐♦♥ ♦❢ t❤❡ ●■

• r❛♣❤ ■s♦♠♦r♣❤✐s♠ ♣r♦❜❧❡♠

◮ ●■✲♣r♦❜❧❡♠✿ ❉❡❝✐❞❡ ✇❤❡t❤❡r t✇♦ ❣✐✈❡♥ ❣r❛♣❤s G, H ∈ Gn ❛r❡

✐s♦♠♦r♣❤✐❝✱ ✐✳❡✳ ❛r❡ ✐♥ t❤❡ s❛♠❡ ♦r❜✐t ♦❢ t❤❡ ❛❝t✐♦♥ Sn Gn✳ ❖✉tst❛♥❞✐♥❣ ♣❛rt✐❝✉❧❛r ❝❛s❡✿ ■s ●■ ✐♥ t❤❡ ❝❧❛ss ❄ ❞❡♥♦t❡s t❤❡ ❝❧❛ss ♦❢ ♣r♦❜❧❡♠ t❤❛t ❤❛✈❡ ❞❡❝✐s✐♦♥ ❛❧❣♦r✐t❤♠s ✇✐t❤ r✉♥♥✐♥❣ t✐♠❡ ❜♦✉♥❞❡❞ ❜② s♦♠❡ ♣♦❧②♥♦♠✐❛❧ ✭✐♥ t❤❡ ✐♥♣✉t ❧❡♥❣t❤✮✳

❙❡❛r❝❤✐♥❣ ❢♦r ❛ ●r❛♣❤ ■s♦♠♦r♣❤✐s♠ ❋♦r♠✉❧❛t✐♦♥ ♦❢ t❤❡ ●■

• r❛♣❤ ■s♦♠♦r♣❤✐s♠ ♣r♦❜❧❡♠

◮ ●■✲♣r♦❜❧❡♠✿ ❉❡❝✐❞❡ ✇❤❡t❤❡r t✇♦ ❣✐✈❡♥ ❣r❛♣❤s G, H ∈ Gn ❛r❡

✐s♦♠♦r♣❤✐❝✱ ✐✳❡✳ ❛r❡ ✐♥ t❤❡ s❛♠❡ ♦r❜✐t ♦❢ t❤❡ ❛❝t✐♦♥ Sn Gn✳

◮ ❖✉tst❛♥❞✐♥❣ ♣❛rt✐❝✉❧❛r ❝❛s❡✿ ■s ●■ ✐♥ t❤❡ ❝❧❛ss P❄

❞❡♥♦t❡s t❤❡ ❝❧❛ss ♦❢ ♣r♦❜❧❡♠ t❤❛t ❤❛✈❡ ❞❡❝✐s✐♦♥ ❛❧❣♦r✐t❤♠s ✇✐t❤ r✉♥♥✐♥❣ t✐♠❡ ❜♦✉♥❞❡❞ ❜② s♦♠❡ ♣♦❧②♥♦♠✐❛❧ ✭✐♥ t❤❡ ✐♥♣✉t ❧❡♥❣t❤✮✳

❙❡❛r❝❤✐♥❣ ❢♦r ❛ ●r❛♣❤ ■s♦♠♦r♣❤✐s♠ ❋♦r♠✉❧❛t✐♦♥ ♦❢ t❤❡ ●■

• r❛♣❤ ■s♦♠♦r♣❤✐s♠ ♣r♦❜❧❡♠

◮ ●■✲♣r♦❜❧❡♠✿ ❉❡❝✐❞❡ ✇❤❡t❤❡r t✇♦ ❣✐✈❡♥ ❣r❛♣❤s G, H ∈ Gn ❛r❡

✐s♦♠♦r♣❤✐❝✱ ✐✳❡✳ ❛r❡ ✐♥ t❤❡ s❛♠❡ ♦r❜✐t ♦❢ t❤❡ ❛❝t✐♦♥ Sn Gn✳

◮ ❖✉tst❛♥❞✐♥❣ ♣❛rt✐❝✉❧❛r ❝❛s❡✿ ■s ●■ ✐♥ t❤❡ ❝❧❛ss P❄ ◮ P ❞❡♥♦t❡s t❤❡ ❝❧❛ss ♦❢ ♣r♦❜❧❡♠ t❤❛t ❤❛✈❡ ❞❡❝✐s✐♦♥ ❛❧❣♦r✐t❤♠s

✇✐t❤ r✉♥♥✐♥❣ t✐♠❡ ❜♦✉♥❞❡❞ ❜② s♦♠❡ ♣♦❧②♥♦♠✐❛❧ ✭✐♥ t❤❡ ✐♥♣✉t ❧❡♥❣t❤✮✳

❙❡❛r❝❤✐♥❣ ❢♦r ❛ ●r❛♣❤ ■s♦♠♦r♣❤✐s♠ ❋♦r♠✉❧❛t✐♦♥ ♦❢ t❤❡ ●■

• r❛♣❤ ■s♦♠♦r♣❤✐s♠ ♣r♦❜❧❡♠

◮ ●■✲♣r♦❜❧❡♠✿ ❉❡❝✐❞❡ ✇❤❡t❤❡r t✇♦ ❣✐✈❡♥ ❣r❛♣❤s G, H ∈ Gn ❛r❡

✐s♦♠♦r♣❤✐❝✱ ✐✳❡✳ ❛r❡ ✐♥ t❤❡ s❛♠❡ ♦r❜✐t ♦❢ t❤❡ ❛❝t✐♦♥ Sn Gn✳

◮ ❖✉tst❛♥❞✐♥❣ ♣❛rt✐❝✉❧❛r ❝❛s❡✿ ■s ●■ ✐♥ t❤❡ ❝❧❛ss P❄ ◮ P ❞❡♥♦t❡s t❤❡ ❝❧❛ss ♦❢ ♣r♦❜❧❡♠ t❤❛t ❤❛✈❡ ❞❡❝✐s✐♦♥ ❛❧❣♦r✐t❤♠s

✇✐t❤ r✉♥♥✐♥❣ t✐♠❡ ❜♦✉♥❞❡❞ ❜② s♦♠❡ ♣♦❧②♥♦♠✐❛❧ ✭✐♥ t❤❡ ✐♥♣✉t ❧❡♥❣t❤✮✳

❙❡❛r❝❤✐♥❣ ❢♦r ❛ ●r❛♣❤ ■s♦♠♦r♣❤✐s♠ ❋♦r♠✉❧❛t✐♦♥ ♦❢ t❤❡ ●■

◮ ▼❛t❤❙❝✐◆❡t✿ P✉❜❧✐❝❛t✐♦♥s r❡s✉❧ts ❢♦r ✧❆♥②✇❤❡r❡❂✭●r❛♣❤

✐s♦♠♦r♣❤✐s♠✮✲ ✺✺✹✳ ❋✐rst ♣✉❜❧✐❝❛t✐♦♥✿ ❈♦r♥❡✐❧✱ ❉❡r❡❦ ●♦r❞♦♥ ●❘❆P❍ ■❙❖▼❖❘P❍■❙▼✳ ❚❤❡s✐s ✭P❤✳❉✳✮✲❯♥✐✈❡rs✐t② ♦❢ ❚♦r♦♥t♦ ✭❈❛♥❛❞❛✮✳ ✶✾✻✽✳

❙❡❛r❝❤✐♥❣ ❢♦r ❛ ●r❛♣❤ ■s♦♠♦r♣❤✐s♠ ❋♦r♠✉❧❛t✐♦♥ ♦❢ t❤❡ ●■

◮ ▼❛t❤❙❝✐◆❡t✿ P✉❜❧✐❝❛t✐♦♥s r❡s✉❧ts ❢♦r ✧❆♥②✇❤❡r❡❂✭●r❛♣❤

✐s♦♠♦r♣❤✐s♠✮✲ ✺✺✹✳

◮ ❋✐rst ♣✉❜❧✐❝❛t✐♦♥✿

❈♦r♥❡✐❧✱ ❉❡r❡❦ ●♦r❞♦♥ ●❘❆P❍ ■❙❖▼❖❘P❍■❙▼✳ ❚❤❡s✐s ✭P❤✳❉✳✮✲❯♥✐✈❡rs✐t② ♦❢ ❚♦r♦♥t♦ ✭❈❛♥❛❞❛✮✳ ✶✾✻✽✳

❙❡❛r❝❤✐♥❣ ❢♦r ❛ ●r❛♣❤ ■s♦♠♦r♣❤✐s♠ ❋♦r♠✉❧❛t✐♦♥ ♦❢ t❤❡ ●■

◮ ▼❛t❤❙❝✐◆❡t✿ P✉❜❧✐❝❛t✐♦♥s r❡s✉❧ts ❢♦r ✧❆♥②✇❤❡r❡❂✭●r❛♣❤

✐s♦♠♦r♣❤✐s♠✮✲ ✺✺✹✳

◮ ❋✐rst ♣✉❜❧✐❝❛t✐♦♥✿ ◮ ❈♦r♥❡✐❧✱ ❉❡r❡❦ ●♦r❞♦♥ ●❘❆P❍ ■❙❖▼❖❘P❍■❙▼✳ ❚❤❡s✐s

✭P❤✳❉✳✮✲❯♥✐✈❡rs✐t② ♦❢ ❚♦r♦♥t♦ ✭❈❛♥❛❞❛✮✳ ✶✾✻✽✳

❙❡❛r❝❤✐♥❣ ❢♦r ❛ ●r❛♣❤ ■s♦♠♦r♣❤✐s♠ ❋♦r♠✉❧❛t✐♦♥ ♦❢ t❤❡ ●■

◮ ▼❛t❤❙❝✐◆❡t✿ P✉❜❧✐❝❛t✐♦♥s r❡s✉❧ts ❢♦r ✧❆♥②✇❤❡r❡❂✭●r❛♣❤

✐s♦♠♦r♣❤✐s♠✮✲ ✺✺✹✳

◮ ❋✐rst ♣✉❜❧✐❝❛t✐♦♥✿ ◮ ❈♦r♥❡✐❧✱ ❉❡r❡❦ ●♦r❞♦♥ ●❘❆P❍ ■❙❖▼❖❘P❍■❙▼✳ ❚❤❡s✐s

✭P❤✳❉✳✮✲❯♥✐✈❡rs✐t② ♦❢ ❚♦r♦♥t♦ ✭❈❛♥❛❞❛✮✳ ✶✾✻✽✳

❙❡❛r❝❤✐♥❣ ❢♦r ❛ ●r❛♣❤ ■s♦♠♦r♣❤✐s♠ ❙❡❛r❝❤✐♥❣ ❢♦r ❛ ●■

❙●■

◮ ❇♦♦t❤✱ ❑❡❧❧♦❣❣ ❙✳❀ ❈♦❧❜♦✉r♥✱ ❈✳ ❏✳ ✭✶✾✼✼✮✱ Pr♦❜❧❡♠s

♣♦❧②♥♦♠✐❛❧❧② ❡q✉✐✈❛❧❡♥t t♦ ❣r❛♣❤ ✐s♦♠♦r♣❤✐s♠✳ ✧✳✳✳ ❚❤❡ ❣r❛♣❤ ✐s♦♠♦r♣❤✐s♠ ✐s t♦ ❞❡❝✐❞❡✱ ❣✐✈❡♥ t✇♦ ❣r❛♣❤s✱ ✇❤❡t❤❡r ❛♥ ✐s♦♠♦r♣❤✐s♠ ❡①✐sts✳ ❆ r❡❧❛t❡❞ ♣r♦❜❧❡♠ ✐s t♦ ✜♥❞ ❛♥ ✐s♦♠♦r♣❤✐s♠ ❛♥❞ ❡①♣❧✐❝✐t❧② ♣r❡s❡♥t ✐t✱ ✇❤❡♥❡✈❡r ♦♥❡ ❡①✐sts✳ ❚❤❡ ❡q✉✐✈❛❧❡♥❝❡ ♦❢ t❤❡s❡ t✇♦ ♣r♦❜❧❡♠ ❤❛s ❜❡❡♥ ❦♥♦✇♥ ❢♦r ♠❛♥② ②❡❛rs✳✳✳✧ ❈✳▼✳ ❍♦✛♠❛♥♥✳ ●r♦✉♣✲❚❤❡♦r❡t✐❝ ❆❧❣♦r✐t❤♠s ❛♥❞ ●r❛♣❤ ■s♦♠♦r♣❤✐s♠✳ ▲◆❈❙ ✶✸✻✱ ❙♣r✐♥❣❡r✲❱❡r❧❛❣ ✶✾✽✷✳ P✶✿ ●✐✈❡♥ t✇♦ ❣r❛♣❤s ❛♥❞ ✇✐t❤ ✈❡rt✐❝❡s ❡❛❝❤✱ ❞❡❝✐❞❡ ✇❤❡t❤❡r t❤❡② ❛r❡ ✐s♦♠♦r♣❤✐❝✳ P✷✿ ●✐✈❡♥ t✇♦ ❣r❛♣❤s ❛♥❞ ✱ ❞❡❝✐❞❡ ✇❤❡t❤❡r t❤❡② ❛r❡ ✐s♦♠♦r♣❤✐❝✱ ❛♥❞ ✐❢ s♦✱ ❝♦♥str✉❝t ❛♥ ✐s♦♠♦r♣❤✐s♠ ❢r♦♠ t♦ ✳

❙❡❛r❝❤✐♥❣ ❢♦r ❛ ●r❛♣❤ ■s♦♠♦r♣❤✐s♠ ❙❡❛r❝❤✐♥❣ ❢♦r ❛ ●■

❙●■

◮ ❇♦♦t❤✱ ❑❡❧❧♦❣❣ ❙✳❀ ❈♦❧❜♦✉r♥✱ ❈✳ ❏✳ ✭✶✾✼✼✮✱ Pr♦❜❧❡♠s

♣♦❧②♥♦♠✐❛❧❧② ❡q✉✐✈❛❧❡♥t t♦ ❣r❛♣❤ ✐s♦♠♦r♣❤✐s♠✳

◮ ✧✳✳✳ ❚❤❡ ❣r❛♣❤ ✐s♦♠♦r♣❤✐s♠ ✐s t♦ ❞❡❝✐❞❡✱ ❣✐✈❡♥ t✇♦ ❣r❛♣❤s✱

✇❤❡t❤❡r ❛♥ ✐s♦♠♦r♣❤✐s♠ ❡①✐sts✳ ❆ r❡❧❛t❡❞ ♣r♦❜❧❡♠ ✐s t♦ ✜♥❞ ❛♥ ✐s♦♠♦r♣❤✐s♠ ❛♥❞ ❡①♣❧✐❝✐t❧② ♣r❡s❡♥t ✐t✱ ✇❤❡♥❡✈❡r ♦♥❡ ❡①✐sts✳ ❚❤❡ ❡q✉✐✈❛❧❡♥❝❡ ♦❢ t❤❡s❡ t✇♦ ♣r♦❜❧❡♠ ❤❛s ❜❡❡♥ ❦♥♦✇♥ ❢♦r ♠❛♥② ②❡❛rs✳✳✳✧ ❈✳▼✳ ❍♦✛♠❛♥♥✳ ●r♦✉♣✲❚❤❡♦r❡t✐❝ ❆❧❣♦r✐t❤♠s ❛♥❞ ●r❛♣❤ ■s♦♠♦r♣❤✐s♠✳ ▲◆❈❙ ✶✸✻✱ ❙♣r✐♥❣❡r✲❱❡r❧❛❣ ✶✾✽✷✳ P✶✿ ●✐✈❡♥ t✇♦ ❣r❛♣❤s ❛♥❞ ✇✐t❤ ✈❡rt✐❝❡s ❡❛❝❤✱ ❞❡❝✐❞❡ ✇❤❡t❤❡r t❤❡② ❛r❡ ✐s♦♠♦r♣❤✐❝✳ P✷✿ ●✐✈❡♥ t✇♦ ❣r❛♣❤s ❛♥❞ ✱ ❞❡❝✐❞❡ ✇❤❡t❤❡r t❤❡② ❛r❡ ✐s♦♠♦r♣❤✐❝✱ ❛♥❞ ✐❢ s♦✱ ❝♦♥str✉❝t ❛♥ ✐s♦♠♦r♣❤✐s♠ ❢r♦♠ t♦ ✳

❙❡❛r❝❤✐♥❣ ❢♦r ❛ ●r❛♣❤ ■s♦♠♦r♣❤✐s♠ ❙❡❛r❝❤✐♥❣ ❢♦r ❛ ●■

❙●■

◮ ❇♦♦t❤✱ ❑❡❧❧♦❣❣ ❙✳❀ ❈♦❧❜♦✉r♥✱ ❈✳ ❏✳ ✭✶✾✼✼✮✱ Pr♦❜❧❡♠s

♣♦❧②♥♦♠✐❛❧❧② ❡q✉✐✈❛❧❡♥t t♦ ❣r❛♣❤ ✐s♦♠♦r♣❤✐s♠✳

◮ ✧✳✳✳ ❚❤❡ ❣r❛♣❤ ✐s♦♠♦r♣❤✐s♠ ✐s t♦ ❞❡❝✐❞❡✱ ❣✐✈❡♥ t✇♦ ❣r❛♣❤s✱

✇❤❡t❤❡r ❛♥ ✐s♦♠♦r♣❤✐s♠ ❡①✐sts✳ ❆ r❡❧❛t❡❞ ♣r♦❜❧❡♠ ✐s t♦ ✜♥❞ ❛♥ ✐s♦♠♦r♣❤✐s♠ ❛♥❞ ❡①♣❧✐❝✐t❧② ♣r❡s❡♥t ✐t✱ ✇❤❡♥❡✈❡r ♦♥❡ ❡①✐sts✳ ❚❤❡ ❡q✉✐✈❛❧❡♥❝❡ ♦❢ t❤❡s❡ t✇♦ ♣r♦❜❧❡♠ ❤❛s ❜❡❡♥ ❦♥♦✇♥ ❢♦r ♠❛♥② ②❡❛rs✳✳✳✧

◮ ❈✳▼✳ ❍♦✛♠❛♥♥✳ ●r♦✉♣✲❚❤❡♦r❡t✐❝ ❆❧❣♦r✐t❤♠s ❛♥❞ ●r❛♣❤

■s♦♠♦r♣❤✐s♠✳ ▲◆❈❙ ✶✸✻✱ ❙♣r✐♥❣❡r✲❱❡r❧❛❣ ✶✾✽✷✳ P✶✿ ●✐✈❡♥ t✇♦ ❣r❛♣❤s ❛♥❞ ✇✐t❤ ✈❡rt✐❝❡s ❡❛❝❤✱ ❞❡❝✐❞❡ ✇❤❡t❤❡r t❤❡② ❛r❡ ✐s♦♠♦r♣❤✐❝✳ P✷✿ ●✐✈❡♥ t✇♦ ❣r❛♣❤s ❛♥❞ ✱ ❞❡❝✐❞❡ ✇❤❡t❤❡r t❤❡② ❛r❡ ✐s♦♠♦r♣❤✐❝✱ ❛♥❞ ✐❢ s♦✱ ❝♦♥str✉❝t ❛♥ ✐s♦♠♦r♣❤✐s♠ ❢r♦♠ t♦ ✳

❙❡❛r❝❤✐♥❣ ❢♦r ❛ ●r❛♣❤ ■s♦♠♦r♣❤✐s♠ ❙❡❛r❝❤✐♥❣ ❢♦r ❛ ●■

❙●■

◮ ❇♦♦t❤✱ ❑❡❧❧♦❣❣ ❙✳❀ ❈♦❧❜♦✉r♥✱ ❈✳ ❏✳ ✭✶✾✼✼✮✱ Pr♦❜❧❡♠s

♣♦❧②♥♦♠✐❛❧❧② ❡q✉✐✈❛❧❡♥t t♦ ❣r❛♣❤ ✐s♦♠♦r♣❤✐s♠✳

◮ ✧✳✳✳ ❚❤❡ ❣r❛♣❤ ✐s♦♠♦r♣❤✐s♠ ✐s t♦ ❞❡❝✐❞❡✱ ❣✐✈❡♥ t✇♦ ❣r❛♣❤s✱

✇❤❡t❤❡r ❛♥ ✐s♦♠♦r♣❤✐s♠ ❡①✐sts✳ ❆ r❡❧❛t❡❞ ♣r♦❜❧❡♠ ✐s t♦ ✜♥❞ ❛♥ ✐s♦♠♦r♣❤✐s♠ ❛♥❞ ❡①♣❧✐❝✐t❧② ♣r❡s❡♥t ✐t✱ ✇❤❡♥❡✈❡r ♦♥❡ ❡①✐sts✳ ❚❤❡ ❡q✉✐✈❛❧❡♥❝❡ ♦❢ t❤❡s❡ t✇♦ ♣r♦❜❧❡♠ ❤❛s ❜❡❡♥ ❦♥♦✇♥ ❢♦r ♠❛♥② ②❡❛rs✳✳✳✧

◮ ❈✳▼✳ ❍♦✛♠❛♥♥✳ ●r♦✉♣✲❚❤❡♦r❡t✐❝ ❆❧❣♦r✐t❤♠s ❛♥❞ ●r❛♣❤

■s♦♠♦r♣❤✐s♠✳ ▲◆❈❙ ✶✸✻✱ ❙♣r✐♥❣❡r✲❱❡r❧❛❣ ✶✾✽✷✳

◮ P✶✿ ●✐✈❡♥ t✇♦ ❣r❛♣❤s G ❛♥❞ H ✇✐t❤ n ✈❡rt✐❝❡s ❡❛❝❤✱ ❞❡❝✐❞❡

✇❤❡t❤❡r t❤❡② ❛r❡ ✐s♦♠♦r♣❤✐❝✳ P✷✿ ●✐✈❡♥ t✇♦ ❣r❛♣❤s G ❛♥❞ H✱ ❞❡❝✐❞❡ ✇❤❡t❤❡r t❤❡② ❛r❡ ✐s♦♠♦r♣❤✐❝✱ ❛♥❞ ✐❢ s♦✱ ❝♦♥str✉❝t ❛♥ ✐s♦♠♦r♣❤✐s♠ ❢r♦♠ G t♦ H✳

❙❡❛r❝❤✐♥❣ ❢♦r ❛ ●r❛♣❤ ■s♦♠♦r♣❤✐s♠ ❙❡❛r❝❤✐♥❣ ❢♦r ❛ ●■

❙●■

◮ ❇♦♦t❤✱ ❑❡❧❧♦❣❣ ❙✳❀ ❈♦❧❜♦✉r♥✱ ❈✳ ❏✳ ✭✶✾✼✼✮✱ Pr♦❜❧❡♠s

♣♦❧②♥♦♠✐❛❧❧② ❡q✉✐✈❛❧❡♥t t♦ ❣r❛♣❤ ✐s♦♠♦r♣❤✐s♠✳

◮ ✧✳✳✳ ❚❤❡ ❣r❛♣❤ ✐s♦♠♦r♣❤✐s♠ ✐s t♦ ❞❡❝✐❞❡✱ ❣✐✈❡♥ t✇♦ ❣r❛♣❤s✱

✇❤❡t❤❡r ❛♥ ✐s♦♠♦r♣❤✐s♠ ❡①✐sts✳ ❆ r❡❧❛t❡❞ ♣r♦❜❧❡♠ ✐s t♦ ✜♥❞ ❛♥ ✐s♦♠♦r♣❤✐s♠ ❛♥❞ ❡①♣❧✐❝✐t❧② ♣r❡s❡♥t ✐t✱ ✇❤❡♥❡✈❡r ♦♥❡ ❡①✐sts✳ ❚❤❡ ❡q✉✐✈❛❧❡♥❝❡ ♦❢ t❤❡s❡ t✇♦ ♣r♦❜❧❡♠ ❤❛s ❜❡❡♥ ❦♥♦✇♥ ❢♦r ♠❛♥② ②❡❛rs✳✳✳✧

◮ ❈✳▼✳ ❍♦✛♠❛♥♥✳ ●r♦✉♣✲❚❤❡♦r❡t✐❝ ❆❧❣♦r✐t❤♠s ❛♥❞ ●r❛♣❤

■s♦♠♦r♣❤✐s♠✳ ▲◆❈❙ ✶✸✻✱ ❙♣r✐♥❣❡r✲❱❡r❧❛❣ ✶✾✽✷✳

◮ P✶✿ ●✐✈❡♥ t✇♦ ❣r❛♣❤s G ❛♥❞ H ✇✐t❤ n ✈❡rt✐❝❡s ❡❛❝❤✱ ❞❡❝✐❞❡

✇❤❡t❤❡r t❤❡② ❛r❡ ✐s♦♠♦r♣❤✐❝✳ P✷✿ ●✐✈❡♥ t✇♦ ❣r❛♣❤s G ❛♥❞ H✱ ❞❡❝✐❞❡ ✇❤❡t❤❡r t❤❡② ❛r❡ ✐s♦♠♦r♣❤✐❝✱ ❛♥❞ ✐❢ s♦✱ ❝♦♥str✉❝t ❛♥ ✐s♦♠♦r♣❤✐s♠ ❢r♦♠ G t♦ H✳

◮ P1 ≡pol P2!

❙❡❛r❝❤✐♥❣ ❢♦r ❛ ●r❛♣❤ ■s♦♠♦r♣❤✐s♠ ❙❡❛r❝❤✐♥❣ ❢♦r ❛ ●■

❙●■

◮ ❇♦♦t❤✱ ❑❡❧❧♦❣❣ ❙✳❀ ❈♦❧❜♦✉r♥✱ ❈✳ ❏✳ ✭✶✾✼✼✮✱ Pr♦❜❧❡♠s

♣♦❧②♥♦♠✐❛❧❧② ❡q✉✐✈❛❧❡♥t t♦ ❣r❛♣❤ ✐s♦♠♦r♣❤✐s♠✳

◮ ✧✳✳✳ ❚❤❡ ❣r❛♣❤ ✐s♦♠♦r♣❤✐s♠ ✐s t♦ ❞❡❝✐❞❡✱ ❣✐✈❡♥ t✇♦ ❣r❛♣❤s✱

✇❤❡t❤❡r ❛♥ ✐s♦♠♦r♣❤✐s♠ ❡①✐sts✳ ❆ r❡❧❛t❡❞ ♣r♦❜❧❡♠ ✐s t♦ ✜♥❞ ❛♥ ✐s♦♠♦r♣❤✐s♠ ❛♥❞ ❡①♣❧✐❝✐t❧② ♣r❡s❡♥t ✐t✱ ✇❤❡♥❡✈❡r ♦♥❡ ❡①✐sts✳ ❚❤❡ ❡q✉✐✈❛❧❡♥❝❡ ♦❢ t❤❡s❡ t✇♦ ♣r♦❜❧❡♠ ❤❛s ❜❡❡♥ ❦♥♦✇♥ ❢♦r ♠❛♥② ②❡❛rs✳✳✳✧

◮ ❈✳▼✳ ❍♦✛♠❛♥♥✳ ●r♦✉♣✲❚❤❡♦r❡t✐❝ ❆❧❣♦r✐t❤♠s ❛♥❞ ●r❛♣❤

■s♦♠♦r♣❤✐s♠✳ ▲◆❈❙ ✶✸✻✱ ❙♣r✐♥❣❡r✲❱❡r❧❛❣ ✶✾✽✷✳

◮ P✶✿ ●✐✈❡♥ t✇♦ ❣r❛♣❤s G ❛♥❞ H ✇✐t❤ n ✈❡rt✐❝❡s ❡❛❝❤✱ ❞❡❝✐❞❡

✇❤❡t❤❡r t❤❡② ❛r❡ ✐s♦♠♦r♣❤✐❝✳ P✷✿ ●✐✈❡♥ t✇♦ ❣r❛♣❤s G ❛♥❞ H✱ ❞❡❝✐❞❡ ✇❤❡t❤❡r t❤❡② ❛r❡ ✐s♦♠♦r♣❤✐❝✱ ❛♥❞ ✐❢ s♦✱ ❝♦♥str✉❝t ❛♥ ✐s♦♠♦r♣❤✐s♠ ❢r♦♠ G t♦ H✳

◮ P1 ≡pol P2!

❙❡❛r❝❤✐♥❣ ❢♦r ❛ ●r❛♣❤ ■s♦♠♦r♣❤✐s♠ ❙❡❛r❝❤✐♥❣ ❢♦r ❛ ●■

❙●■

◮

Ðèñ✳✿ ❆✳◆✳❘②❜❛❧♦✈✱ st❛t✐♥❣ ❙●■

❚❤❡ ✐♥♣✉t s❡t ❢♦r s❡❛r❝❤✐♥❣ ❣r❛♣❤ ✐s♦♠♦r♣❤✐s♠ ♣r♦❜❧❡♠ ❙●■ ✐s t❤❡ s❡t ♦❢ ❛❧❧ ♣❛✐rs ♦❢ ✐s♦♠♦r♣❤✐❝ ❣r❛♣❤s✳

• ✐✈❡♥ t✇♦ ✐s♦♠♦r♣❤✐❝ ❣r❛♣❤s

❛♥❞ ✱ ♦♥❡ ♥❡❡❞s t♦ ❝♦♥str✉❝t ❛♥ ✐s♦♠♦r♣❤✐s♠ ❜❡t✇❡❡♥ ❛♥❞ ✳

❙❡❛r❝❤✐♥❣ ❢♦r ❛ ●r❛♣❤ ■s♦♠♦r♣❤✐s♠ ❙❡❛r❝❤✐♥❣ ❢♦r ❛ ●■

❙●■

◮

Ðèñ✳✿ ❆✳◆✳❘②❜❛❧♦✈✱ st❛t✐♥❣ ❙●■

◮ ❚❤❡ ✐♥♣✉t s❡t ❢♦r s❡❛r❝❤✐♥❣ ❣r❛♣❤ ✐s♦♠♦r♣❤✐s♠ ♣r♦❜❧❡♠ ❙●■ ✐s

t❤❡ s❡t ♦❢ ❛❧❧ ♣❛✐rs ♦❢ ✐s♦♠♦r♣❤✐❝ ❣r❛♣❤s✳

• ✐✈❡♥ t✇♦ ✐s♦♠♦r♣❤✐❝ ❣r❛♣❤s

❛♥❞ ✱ ♦♥❡ ♥❡❡❞s t♦ ❝♦♥str✉❝t ❛♥ ✐s♦♠♦r♣❤✐s♠ ❜❡t✇❡❡♥ ❛♥❞ ✳

❙❡❛r❝❤✐♥❣ ❢♦r ❛ ●r❛♣❤ ■s♦♠♦r♣❤✐s♠ ❙❡❛r❝❤✐♥❣ ❢♦r ❛ ●■

❙●■

◮

Ðèñ✳✿ ❆✳◆✳❘②❜❛❧♦✈✱ st❛t✐♥❣ ❙●■

◮ ❚❤❡ ✐♥♣✉t s❡t ❢♦r s❡❛r❝❤✐♥❣ ❣r❛♣❤ ✐s♦♠♦r♣❤✐s♠ ♣r♦❜❧❡♠ ❙●■ ✐s

t❤❡ s❡t ♦❢ ❛❧❧ ♣❛✐rs ♦❢ ✐s♦♠♦r♣❤✐❝ ❣r❛♣❤s✳

◮ ●✐✈❡♥ t✇♦ ✐s♦♠♦r♣❤✐❝ ❣r❛♣❤s G ❛♥❞ H✱ ♦♥❡ ♥❡❡❞s t♦ ❝♦♥str✉❝t

❛♥ ✐s♦♠♦r♣❤✐s♠ ❜❡t✇❡❡♥ G ❛♥❞ H✳

❙❡❛r❝❤✐♥❣ ❢♦r ❛ ●r❛♣❤ ■s♦♠♦r♣❤✐s♠ ❙❡❛r❝❤✐♥❣ ❢♦r ❛ ●■

❙●■

◮

Ðèñ✳✿ ❆✳◆✳❘②❜❛❧♦✈✱ st❛t✐♥❣ ❙●■

◮ ❚❤❡ ✐♥♣✉t s❡t ❢♦r s❡❛r❝❤✐♥❣ ❣r❛♣❤ ✐s♦♠♦r♣❤✐s♠ ♣r♦❜❧❡♠ ❙●■ ✐s

t❤❡ s❡t ♦❢ ❛❧❧ ♣❛✐rs ♦❢ ✐s♦♠♦r♣❤✐❝ ❣r❛♣❤s✳

◮ ●✐✈❡♥ t✇♦ ✐s♦♠♦r♣❤✐❝ ❣r❛♣❤s G ❛♥❞ H✱ ♦♥❡ ♥❡❡❞s t♦ ❝♦♥str✉❝t

❛♥ ✐s♦♠♦r♣❤✐s♠ ❜❡t✇❡❡♥ G ❛♥❞ H✳

❙❡❛r❝❤✐♥❣ ❢♦r ❛ ●r❛♣❤ ■s♦♠♦r♣❤✐s♠ ❙❡❛r❝❤✐♥❣ ❢♦r ❛ ●■

❙●■ ❛♥❞ ♣r♦♠✐s❡ ♣r♦❜❧❡♠s

◮ Pr♦♠✐s❡ ♣r♦❜❧❡♠s ✭❊✈❡♥✲ ❙❡❧♠❛♥ ✲❨❛❝♦❜✐ ✭✶✾✽✹✮✮

❋♦r ❛ ♣♦❧②♥♦♠✐❛❧❧② ❜♦✉♥❞❡❞ r❡❧❛t✐♦♥ ✱ t❤❡ s❡❛r❝❤ ✭♣r♦♠✐s❡✮ ♣r♦❜❧❡♠ ✐s ❣✐✈❡♥ t❤❛t ❤❛s ❛ s♦❧✉t✐♦♥ ✭✐✳❡✳✱ t❤❡r❡ ❡①✐sts s✉❝❤ t❤❛t t♦ ✜♥❞ s✉❝❤ ❛ s♦❧✉t✐♦♥ ✭✐✳❡✳✱ ✜♥❞ ❛ s✉❝❤ t❤❛t ✮✳ ❍❡♥❝❡✱ t❤❡ ♣r♦♠✐s❡ ✐s t❤❛t ❤❛s ❛ s♦❧✉t✐♦♥ ✭✐✳❡✳✱ ② s✉❝❤ t❤❛t ✱ ❛♥❞ ♥♦t❤✐♥❣ ✐s r❡q✉✐r❡❞ ✐♥ ❝❛s❡ ❤❛s ♥♦ s♦❧✉t✐♦♥✳ ❙●■✿ ✳ ●✐✈❡♥ s✉❝❤ t❤❛t t♦ ✜♥❞

❙❡❛r❝❤✐♥❣ ❢♦r ❛ ●r❛♣❤ ■s♦♠♦r♣❤✐s♠ ❙❡❛r❝❤✐♥❣ ❢♦r ❛ ●■

❙●■ ❛♥❞ ♣r♦♠✐s❡ ♣r♦❜❧❡♠s

◮ Pr♦♠✐s❡ ♣r♦❜❧❡♠s ✭❊✈❡♥✲ ❙❡❧♠❛♥ ✲❨❛❝♦❜✐ ✭✶✾✽✹✮✮ ◮ ❋♦r ❛ ♣♦❧②♥♦♠✐❛❧❧② ❜♦✉♥❞❡❞ r❡❧❛t✐♦♥ R✱ t❤❡ s❡❛r❝❤ ✭♣r♦♠✐s❡✮

♣r♦❜❧❡♠ ✐s ❣✐✈❡♥ x t❤❛t ❤❛s ❛ s♦❧✉t✐♦♥ ✭✐✳❡✳✱ t❤❡r❡ ❡①✐sts y s✉❝❤ t❤❛t (x, y) ∈ R) t♦ ✜♥❞ s✉❝❤ ❛ s♦❧✉t✐♦♥ ✭✐✳❡✳✱ ✜♥❞ ❛ y s✉❝❤ t❤❛t (x, y) ∈ R)✮✳ ❍❡♥❝❡✱ t❤❡ ♣r♦♠✐s❡ ✐s t❤❛t ❤❛s ❛ s♦❧✉t✐♦♥ ✭✐✳❡✳✱ ② s✉❝❤ t❤❛t ✱ ❛♥❞ ♥♦t❤✐♥❣ ✐s r❡q✉✐r❡❞ ✐♥ ❝❛s❡ ❤❛s ♥♦ s♦❧✉t✐♦♥✳ ❙●■✿ ✳ ●✐✈❡♥ s✉❝❤ t❤❛t t♦ ✜♥❞

❙❡❛r❝❤✐♥❣ ❢♦r ❛ ●r❛♣❤ ■s♦♠♦r♣❤✐s♠ ❙❡❛r❝❤✐♥❣ ❢♦r ❛ ●■

❙●■ ❛♥❞ ♣r♦♠✐s❡ ♣r♦❜❧❡♠s

◮ Pr♦♠✐s❡ ♣r♦❜❧❡♠s ✭❊✈❡♥✲ ❙❡❧♠❛♥ ✲❨❛❝♦❜✐ ✭✶✾✽✹✮✮ ◮ ❋♦r ❛ ♣♦❧②♥♦♠✐❛❧❧② ❜♦✉♥❞❡❞ r❡❧❛t✐♦♥ R✱ t❤❡ s❡❛r❝❤ ✭♣r♦♠✐s❡✮

♣r♦❜❧❡♠ ✐s ❣✐✈❡♥ x t❤❛t ❤❛s ❛ s♦❧✉t✐♦♥ ✭✐✳❡✳✱ t❤❡r❡ ❡①✐sts y s✉❝❤ t❤❛t (x, y) ∈ R) t♦ ✜♥❞ s✉❝❤ ❛ s♦❧✉t✐♦♥ ✭✐✳❡✳✱ ✜♥❞ ❛ y s✉❝❤ t❤❛t (x, y) ∈ R)✮✳

◮ ❍❡♥❝❡✱ t❤❡ ♣r♦♠✐s❡ ✐s t❤❛t x ❤❛s ❛ s♦❧✉t✐♦♥ ✭✐✳❡✳✱ ② s✉❝❤ t❤❛t

(x, y) ∈ R)✱ ❛♥❞ ♥♦t❤✐♥❣ ✐s r❡q✉✐r❡❞ ✐♥ ❝❛s❡ x ❤❛s ♥♦ s♦❧✉t✐♦♥✳ ❙●■✿ ✳ ●✐✈❡♥ s✉❝❤ t❤❛t t♦ ✜♥❞

❙❡❛r❝❤✐♥❣ ❢♦r ❛ ●r❛♣❤ ■s♦♠♦r♣❤✐s♠ ❙❡❛r❝❤✐♥❣ ❢♦r ❛ ●■

❙●■ ❛♥❞ ♣r♦♠✐s❡ ♣r♦❜❧❡♠s

◮ Pr♦♠✐s❡ ♣r♦❜❧❡♠s ✭❊✈❡♥✲ ❙❡❧♠❛♥ ✲❨❛❝♦❜✐ ✭✶✾✽✹✮✮ ◮ ❋♦r ❛ ♣♦❧②♥♦♠✐❛❧❧② ❜♦✉♥❞❡❞ r❡❧❛t✐♦♥ R✱ t❤❡ s❡❛r❝❤ ✭♣r♦♠✐s❡✮

♣r♦❜❧❡♠ ✐s ❣✐✈❡♥ x t❤❛t ❤❛s ❛ s♦❧✉t✐♦♥ ✭✐✳❡✳✱ t❤❡r❡ ❡①✐sts y s✉❝❤ t❤❛t (x, y) ∈ R) t♦ ✜♥❞ s✉❝❤ ❛ s♦❧✉t✐♦♥ ✭✐✳❡✳✱ ✜♥❞ ❛ y s✉❝❤ t❤❛t (x, y) ∈ R)✮✳

◮ ❍❡♥❝❡✱ t❤❡ ♣r♦♠✐s❡ ✐s t❤❛t x ❤❛s ❛ s♦❧✉t✐♦♥ ✭✐✳❡✳✱ ② s✉❝❤ t❤❛t

(x, y) ∈ R)✱ ❛♥❞ ♥♦t❤✐♥❣ ✐s r❡q✉✐r❡❞ ✐♥ ❝❛s❡ x ❤❛s ♥♦ s♦❧✉t✐♦♥✳

◮ ❙●■✿ R(G, H, φ) ⇔ φ : G ≃ H✳ ●✐✈❡♥ (G, H) s✉❝❤ t❤❛t

∃φ : G ≃ H t♦ ✜♥❞ φ.

❙❡❛r❝❤✐♥❣ ❢♦r ❛ ●r❛♣❤ ■s♦♠♦r♣❤✐s♠ ❙❡❛r❝❤✐♥❣ ❢♦r ❛ ●■

❙●■ ❛♥❞ ♣r♦♠✐s❡ ♣r♦❜❧❡♠s

◮ Pr♦♠✐s❡ ♣r♦❜❧❡♠s ✭❊✈❡♥✲ ❙❡❧♠❛♥ ✲❨❛❝♦❜✐ ✭✶✾✽✹✮✮ ◮ ❋♦r ❛ ♣♦❧②♥♦♠✐❛❧❧② ❜♦✉♥❞❡❞ r❡❧❛t✐♦♥ R✱ t❤❡ s❡❛r❝❤ ✭♣r♦♠✐s❡✮

♣r♦❜❧❡♠ ✐s ❣✐✈❡♥ x t❤❛t ❤❛s ❛ s♦❧✉t✐♦♥ ✭✐✳❡✳✱ t❤❡r❡ ❡①✐sts y s✉❝❤ t❤❛t (x, y) ∈ R) t♦ ✜♥❞ s✉❝❤ ❛ s♦❧✉t✐♦♥ ✭✐✳❡✳✱ ✜♥❞ ❛ y s✉❝❤ t❤❛t (x, y) ∈ R)✮✳

◮ ❍❡♥❝❡✱ t❤❡ ♣r♦♠✐s❡ ✐s t❤❛t x ❤❛s ❛ s♦❧✉t✐♦♥ ✭✐✳❡✳✱ ② s✉❝❤ t❤❛t

(x, y) ∈ R)✱ ❛♥❞ ♥♦t❤✐♥❣ ✐s r❡q✉✐r❡❞ ✐♥ ❝❛s❡ x ❤❛s ♥♦ s♦❧✉t✐♦♥✳

◮ ❙●■✿ R(G, H, φ) ⇔ φ : G ≃ H✳ ●✐✈❡♥ (G, H) s✉❝❤ t❤❛t

∃φ : G ≃ H t♦ ✜♥❞ φ.

❙❡❛r❝❤✐♥❣ ❢♦r ❛ ●r❛♣❤ ■s♦♠♦r♣❤✐s♠ ❙●■ ✈❡rs✉s ●■

❙●■ ✐s r❡❞✉❝✐❜❧❡ t♦ ●■

◮ ❚❤❡♦r❡♠

❙●■ ≤pol ●■✳ Pr♦♦❢ ❆s❝❡♥t✳ ❈♦♥str✉❝t ✭✐♥ ♣♦❧②♥♦♠✐❛❧ t✐♠❡✮ ✱ ❛♥❞ s✉❝❤ t❤❛t✿ ✶✮ ❢♦r ❛❧❧ ✱ ✷✮ ❊✈❡r② ✐s♦♠♦r♣❤✐s♠ ❜❡t✇❡❡♥ ❛♥❞ t❛❦❡s t♦ ❢♦r ❛❧❧ ✸✮ ❊✈❡r② ✐s♦♠♦r♣❤✐s♠ ❜❡t✇❡❡♥ ❛♥❞ t❛❦❡s t♦ ❢♦r ❛❧❧ ✳

❙❡❛r❝❤✐♥❣ ❢♦r ❛ ●r❛♣❤ ■s♦♠♦r♣❤✐s♠ ❙●■ ✈❡rs✉s ●■

❙●■ ✐s r❡❞✉❝✐❜❧❡ t♦ ●■

◮ ❚❤❡♦r❡♠

❙●■ ≤pol ●■✳

◮ Pr♦♦❢

❆s❝❡♥t✳ ❈♦♥str✉❝t ✭✐♥ ♣♦❧②♥♦♠✐❛❧ t✐♠❡✮ ✱ ❛♥❞ s✉❝❤ t❤❛t✿ ✶✮ ❢♦r ❛❧❧ ✱ ✷✮ ❊✈❡r② ✐s♦♠♦r♣❤✐s♠ ❜❡t✇❡❡♥ ❛♥❞ t❛❦❡s t♦ ❢♦r ❛❧❧ ✸✮ ❊✈❡r② ✐s♦♠♦r♣❤✐s♠ ❜❡t✇❡❡♥ ❛♥❞ t❛❦❡s t♦ ❢♦r ❛❧❧ ✳

❙❡❛r❝❤✐♥❣ ❢♦r ❛ ●r❛♣❤ ■s♦♠♦r♣❤✐s♠ ❙●■ ✈❡rs✉s ●■

❙●■ ✐s r❡❞✉❝✐❜❧❡ t♦ ●■

◮ ❚❤❡♦r❡♠

❙●■ ≤pol ●■✳

◮ Pr♦♦❢ ◮ ❆s❝❡♥t✳

❈♦♥str✉❝t ✭✐♥ ♣♦❧②♥♦♠✐❛❧ t✐♠❡✮ ✱ ❛♥❞ s✉❝❤ t❤❛t✿ ✶✮ ❢♦r ❛❧❧ ✱ ✷✮ ❊✈❡r② ✐s♦♠♦r♣❤✐s♠ ❜❡t✇❡❡♥ ❛♥❞ t❛❦❡s t♦ ❢♦r ❛❧❧ ✸✮ ❊✈❡r② ✐s♦♠♦r♣❤✐s♠ ❜❡t✇❡❡♥ ❛♥❞ t❛❦❡s t♦ ❢♦r ❛❧❧ ✳

❙❡❛r❝❤✐♥❣ ❢♦r ❛ ●r❛♣❤ ■s♦♠♦r♣❤✐s♠ ❙●■ ✈❡rs✉s ●■

❙●■ ✐s r❡❞✉❝✐❜❧❡ t♦ ●■

◮ ❚❤❡♦r❡♠

❙●■ ≤pol ●■✳

◮ Pr♦♦❢ ◮ ❆s❝❡♥t✳ ◮ VG = {g1, . . . , gn} .

❈♦♥str✉❝t ✭✐♥ ♣♦❧②♥♦♠✐❛❧ t✐♠❡✮ ✱ ❛♥❞ s✉❝❤ t❤❛t✿ ✶✮ ❢♦r ❛❧❧ ✱ ✷✮ ❊✈❡r② ✐s♦♠♦r♣❤✐s♠ ❜❡t✇❡❡♥ ❛♥❞ t❛❦❡s t♦ ❢♦r ❛❧❧ ✸✮ ❊✈❡r② ✐s♦♠♦r♣❤✐s♠ ❜❡t✇❡❡♥ ❛♥❞ t❛❦❡s t♦ ❢♦r ❛❧❧ ✳

❙❡❛r❝❤✐♥❣ ❢♦r ❛ ●r❛♣❤ ■s♦♠♦r♣❤✐s♠ ❙●■ ✈❡rs✉s ●■

❙●■ ✐s r❡❞✉❝✐❜❧❡ t♦ ●■

◮ ❚❤❡♦r❡♠

❙●■ ≤pol ●■✳

◮ Pr♦♦❢ ◮ ❆s❝❡♥t✳ ◮ VG = {g1, . . . , gn} . ◮ ❈♦♥str✉❝t ✭✐♥ ♣♦❧②♥♦♠✐❛❧ t✐♠❡✮ G = G0 < G1 < · · · < Gn✱

H = H0 < H1 < · · · < Hn ❛♥❞ s✉❝❤ t❤❛t✿ ✶✮ ❢♦r ❛❧❧ ✱ ✷✮ ❊✈❡r② ✐s♦♠♦r♣❤✐s♠ ❜❡t✇❡❡♥ ❛♥❞ t❛❦❡s t♦ ❢♦r ❛❧❧ ✸✮ ❊✈❡r② ✐s♦♠♦r♣❤✐s♠ ❜❡t✇❡❡♥ ❛♥❞ t❛❦❡s t♦ ❢♦r ❛❧❧ ✳

❙❡❛r❝❤✐♥❣ ❢♦r ❛ ●r❛♣❤ ■s♦♠♦r♣❤✐s♠ ❙●■ ✈❡rs✉s ●■

❙●■ ✐s r❡❞✉❝✐❜❧❡ t♦ ●■

◮ ❚❤❡♦r❡♠

❙●■ ≤pol ●■✳

◮ Pr♦♦❢ ◮ ❆s❝❡♥t✳ ◮ VG = {g1, . . . , gn} . ◮ ❈♦♥str✉❝t ✭✐♥ ♣♦❧②♥♦♠✐❛❧ t✐♠❡✮ G = G0 < G1 < · · · < Gn✱

H = H0 < H1 < · · · < Hn

◮ ❛♥❞ h1, . . . , hn ∈ VH s✉❝❤ t❤❛t✿

✶✮ ❢♦r ❛❧❧ ✱ ✷✮ ❊✈❡r② ✐s♦♠♦r♣❤✐s♠ ❜❡t✇❡❡♥ ❛♥❞ t❛❦❡s t♦ ❢♦r ❛❧❧ ✸✮ ❊✈❡r② ✐s♦♠♦r♣❤✐s♠ ❜❡t✇❡❡♥ ❛♥❞ t❛❦❡s t♦ ❢♦r ❛❧❧ ✳

❙❡❛r❝❤✐♥❣ ❢♦r ❛ ●r❛♣❤ ■s♦♠♦r♣❤✐s♠ ❙●■ ✈❡rs✉s ●■

❙●■ ✐s r❡❞✉❝✐❜❧❡ t♦ ●■

◮ ❚❤❡♦r❡♠

❙●■ ≤pol ●■✳

◮ Pr♦♦❢ ◮ ❆s❝❡♥t✳ ◮ VG = {g1, . . . , gn} . ◮ ❈♦♥str✉❝t ✭✐♥ ♣♦❧②♥♦♠✐❛❧ t✐♠❡✮ G = G0 < G1 < · · · < Gn✱

H = H0 < H1 < · · · < Hn

◮ ❛♥❞ h1, . . . , hn ∈ VH s✉❝❤ t❤❛t✿ ◮ ✶✮ Gi ≃ Hi ❢♦r ❛❧❧ i✱

✷✮ ❊✈❡r② ✐s♦♠♦r♣❤✐s♠ ❜❡t✇❡❡♥ ❛♥❞ t❛❦❡s t♦ ❢♦r ❛❧❧ ✸✮ ❊✈❡r② ✐s♦♠♦r♣❤✐s♠ ❜❡t✇❡❡♥ ❛♥❞ t❛❦❡s t♦ ❢♦r ❛❧❧ ✳

❙❡❛r❝❤✐♥❣ ❢♦r ❛ ●r❛♣❤ ■s♦♠♦r♣❤✐s♠ ❙●■ ✈❡rs✉s ●■

❙●■ ✐s r❡❞✉❝✐❜❧❡ t♦ ●■

◮ ❚❤❡♦r❡♠

❙●■ ≤pol ●■✳

◮ Pr♦♦❢ ◮ ❆s❝❡♥t✳ ◮ VG = {g1, . . . , gn} . ◮ ❈♦♥str✉❝t ✭✐♥ ♣♦❧②♥♦♠✐❛❧ t✐♠❡✮ G = G0 < G1 < · · · < Gn✱

H = H0 < H1 < · · · < Hn

◮ ❛♥❞ h1, . . . , hn ∈ VH s✉❝❤ t❤❛t✿ ◮ ✶✮ Gi ≃ Hi ❢♦r ❛❧❧ i✱ ◮ ✷✮ ❊✈❡r② ✐s♦♠♦r♣❤✐s♠ ❜❡t✇❡❡♥ Gi ❛♥❞ Hi t❛❦❡s Gi−1 t♦ Hi−1

❢♦r ❛❧❧ i = 1, . . . , n, ✸✮ ❊✈❡r② ✐s♦♠♦r♣❤✐s♠ ❜❡t✇❡❡♥ ❛♥❞ t❛❦❡s t♦ ❢♦r ❛❧❧ ✳

❙❡❛r❝❤✐♥❣ ❢♦r ❛ ●r❛♣❤ ■s♦♠♦r♣❤✐s♠ ❙●■ ✈❡rs✉s ●■

❙●■ ✐s r❡❞✉❝✐❜❧❡ t♦ ●■

◮ ❚❤❡♦r❡♠

❙●■ ≤pol ●■✳

◮ Pr♦♦❢ ◮ ❆s❝❡♥t✳ ◮ VG = {g1, . . . , gn} . ◮ ❈♦♥str✉❝t ✭✐♥ ♣♦❧②♥♦♠✐❛❧ t✐♠❡✮ G = G0 < G1 < · · · < Gn✱

H = H0 < H1 < · · · < Hn

◮ ❛♥❞ h1, . . . , hn ∈ VH s✉❝❤ t❤❛t✿ ◮ ✶✮ Gi ≃ Hi ❢♦r ❛❧❧ i✱ ◮ ✷✮ ❊✈❡r② ✐s♦♠♦r♣❤✐s♠ ❜❡t✇❡❡♥ Gi ❛♥❞ Hi t❛❦❡s Gi−1 t♦ Hi−1

❢♦r ❛❧❧ i = 1, . . . , n,

◮ ✸✮ ❊✈❡r② ✐s♦♠♦r♣❤✐s♠ ❜❡t✇❡❡♥ Gi ❛♥❞ Hi t❛❦❡s gi t♦ hi ❢♦r ❛❧❧

i = 1, . . . , n✳

❙❡❛r❝❤✐♥❣ ❢♦r ❛ ●r❛♣❤ ■s♦♠♦r♣❤✐s♠ ❙●■ ✈❡rs✉s ●■

❙●■ ✐s r❡❞✉❝✐❜❧❡ t♦ ●■

◮ ❚❤❡♦r❡♠

❙●■ ≤pol ●■✳

◮ Pr♦♦❢ ◮ ❆s❝❡♥t✳ ◮ VG = {g1, . . . , gn} . ◮ ❈♦♥str✉❝t ✭✐♥ ♣♦❧②♥♦♠✐❛❧ t✐♠❡✮ G = G0 < G1 < · · · < Gn✱

H = H0 < H1 < · · · < Hn

◮ ❛♥❞ h1, . . . , hn ∈ VH s✉❝❤ t❤❛t✿ ◮ ✶✮ Gi ≃ Hi ❢♦r ❛❧❧ i✱ ◮ ✷✮ ❊✈❡r② ✐s♦♠♦r♣❤✐s♠ ❜❡t✇❡❡♥ Gi ❛♥❞ Hi t❛❦❡s Gi−1 t♦ Hi−1

❢♦r ❛❧❧ i = 1, . . . , n,

◮ ✸✮ ❊✈❡r② ✐s♦♠♦r♣❤✐s♠ ❜❡t✇❡❡♥ Gi ❛♥❞ Hi t❛❦❡s gi t♦ hi ❢♦r ❛❧❧

i = 1, . . . , n✳

❙❡❛r❝❤✐♥❣ ❢♦r ❛ ●r❛♣❤ ■s♦♠♦r♣❤✐s♠ ❙●■ ✈❡rs✉s ●■

❙●■

◮ ❉❡s❝❡♥t

❚❤❡ ♠❛♣ ✐s ❛♥ ✐s♦♠♦r♣❤✐s♠ ❜❡t✇❡❡♥ ❛♥❞ ✦ ■♥❞❡❡❞✱ ❜② ✶✮ ✳ ❇② ✷✮ t❛❦❡s t♦ ✳ t❛❦❡s t♦ ❜② ✷✮ ❛♥❞ t❛❦❡s t♦ ❜② ✸✮✳ ❚❤❡♥ t❛❦❡s t♦ ❛♥❞ ✇❡ ❝❛♥ ❝♦♥t✐♥✉❡ t❤✐s ❞❡s❝❡♥t ♣r♦❝❡ss ✉♥t✐❧ ♦❜t❛✐♥✐♥❣ t❤❛t t❛❦❡s ❡✈❡r② t♦ ✳ ❍❡♥❝❡ t❤❡ ♠❛♣ ✐s ❛♥ ✐s♦♠♦r♣❤✐s♠ ✐♥ q✉❡st✐♦♥✳

❙❡❛r❝❤✐♥❣ ❢♦r ❛ ●r❛♣❤ ■s♦♠♦r♣❤✐s♠ ❙●■ ✈❡rs✉s ●■

❙●■

◮ ❉❡s❝❡♥t ◮ ❚❤❡ ♠❛♣ gi → hi ✐s ❛♥ ✐s♦♠♦r♣❤✐s♠ ❜❡t✇❡❡♥ G ❛♥❞ H✦

■♥❞❡❡❞✱ ❜② ✶✮ ✳ ❇② ✷✮ t❛❦❡s t♦ ✳ t❛❦❡s t♦ ❜② ✷✮ ❛♥❞ t❛❦❡s t♦ ❜② ✸✮✳ ❚❤❡♥ t❛❦❡s t♦ ❛♥❞ ✇❡ ❝❛♥ ❝♦♥t✐♥✉❡ t❤✐s ❞❡s❝❡♥t ♣r♦❝❡ss ✉♥t✐❧ ♦❜t❛✐♥✐♥❣ t❤❛t t❛❦❡s ❡✈❡r② t♦ ✳ ❍❡♥❝❡ t❤❡ ♠❛♣ ✐s ❛♥ ✐s♦♠♦r♣❤✐s♠ ✐♥ q✉❡st✐♦♥✳

❙❡❛r❝❤✐♥❣ ❢♦r ❛ ●r❛♣❤ ■s♦♠♦r♣❤✐s♠ ❙●■ ✈❡rs✉s ●■

❙●■

◮ ❉❡s❝❡♥t ◮ ❚❤❡ ♠❛♣ gi → hi ✐s ❛♥ ✐s♦♠♦r♣❤✐s♠ ❜❡t✇❡❡♥ G ❛♥❞ H✦ ◮ ■♥❞❡❡❞✱ ❜② ✶✮ ∃φ : Gn → Hn✳

❇② ✷✮ t❛❦❡s t♦ ✳ t❛❦❡s t♦ ❜② ✷✮ ❛♥❞ t❛❦❡s t♦ ❜② ✸✮✳ ❚❤❡♥ t❛❦❡s t♦ ❛♥❞ ✇❡ ❝❛♥ ❝♦♥t✐♥✉❡ t❤✐s ❞❡s❝❡♥t ♣r♦❝❡ss ✉♥t✐❧ ♦❜t❛✐♥✐♥❣ t❤❛t t❛❦❡s ❡✈❡r② t♦ ✳ ❍❡♥❝❡ t❤❡ ♠❛♣ ✐s ❛♥ ✐s♦♠♦r♣❤✐s♠ ✐♥ q✉❡st✐♦♥✳

❙❡❛r❝❤✐♥❣ ❢♦r ❛ ●r❛♣❤ ■s♦♠♦r♣❤✐s♠ ❙●■ ✈❡rs✉s ●■

❙●■

◮ ❉❡s❝❡♥t ◮ ❚❤❡ ♠❛♣ gi → hi ✐s ❛♥ ✐s♦♠♦r♣❤✐s♠ ❜❡t✇❡❡♥ G ❛♥❞ H✦ ◮ ■♥❞❡❡❞✱ ❜② ✶✮ ∃φ : Gn → Hn✳ ◮ ❇② ✷✮ φ t❛❦❡s gn t♦ hn✳

t❛❦❡s t♦ ❜② ✷✮ ❛♥❞ t❛❦❡s t♦ ❜② ✸✮✳ ❚❤❡♥ t❛❦❡s t♦ ❛♥❞ ✇❡ ❝❛♥ ❝♦♥t✐♥✉❡ t❤✐s ❞❡s❝❡♥t ♣r♦❝❡ss ✉♥t✐❧ ♦❜t❛✐♥✐♥❣ t❤❛t t❛❦❡s ❡✈❡r② t♦ ✳ ❍❡♥❝❡ t❤❡ ♠❛♣ ✐s ❛♥ ✐s♦♠♦r♣❤✐s♠ ✐♥ q✉❡st✐♦♥✳

❙❡❛r❝❤✐♥❣ ❢♦r ❛ ●r❛♣❤ ■s♦♠♦r♣❤✐s♠ ❙●■ ✈❡rs✉s ●■

❙●■

◮ ❉❡s❝❡♥t ◮ ❚❤❡ ♠❛♣ gi → hi ✐s ❛♥ ✐s♦♠♦r♣❤✐s♠ ❜❡t✇❡❡♥ G ❛♥❞ H✦ ◮ ■♥❞❡❡❞✱ ❜② ✶✮ ∃φ : Gn → Hn✳ ◮ ❇② ✷✮ φ t❛❦❡s gn t♦ hn✳ ◮ φ|Gn−1 t❛❦❡s Gn−1 t♦ Hn−1 ❜② ✷✮ ❛♥❞ t❛❦❡s gn−1 t♦ hn−1 ❜② ✸✮✳

❚❤❡♥ t❛❦❡s t♦ ❛♥❞ ✇❡ ❝❛♥ ❝♦♥t✐♥✉❡ t❤✐s ❞❡s❝❡♥t ♣r♦❝❡ss ✉♥t✐❧ ♦❜t❛✐♥✐♥❣ t❤❛t t❛❦❡s ❡✈❡r② t♦ ✳ ❍❡♥❝❡ t❤❡ ♠❛♣ ✐s ❛♥ ✐s♦♠♦r♣❤✐s♠ ✐♥ q✉❡st✐♦♥✳

❙❡❛r❝❤✐♥❣ ❢♦r ❛ ●r❛♣❤ ■s♦♠♦r♣❤✐s♠ ❙●■ ✈❡rs✉s ●■

❙●■

◮ ❉❡s❝❡♥t ◮ ❚❤❡ ♠❛♣ gi → hi ✐s ❛♥ ✐s♦♠♦r♣❤✐s♠ ❜❡t✇❡❡♥ G ❛♥❞ H✦ ◮ ■♥❞❡❡❞✱ ❜② ✶✮ ∃φ : Gn → Hn✳ ◮ ❇② ✷✮ φ t❛❦❡s gn t♦ hn✳ ◮ φ|Gn−1 t❛❦❡s Gn−1 t♦ Hn−1 ❜② ✷✮ ❛♥❞ t❛❦❡s gn−1 t♦ hn−1 ❜② ✸✮✳ ◮ ❚❤❡♥ φ|Gn−1t❛❦❡s Gn−2 t♦ Hn−2 ❛♥❞ ✇❡ ❝❛♥ ❝♦♥t✐♥✉❡ t❤✐s

❞❡s❝❡♥t ♣r♦❝❡ss ✉♥t✐❧ ♦❜t❛✐♥✐♥❣ t❤❛t φ t❛❦❡s ❡✈❡r② gi t♦ hi✳ ❍❡♥❝❡ t❤❡ ♠❛♣ gi → hi ✐s ❛♥ ✐s♦♠♦r♣❤✐s♠ ✐♥ q✉❡st✐♦♥✳

❙❡❛r❝❤✐♥❣ ❢♦r ❛ ●r❛♣❤ ■s♦♠♦r♣❤✐s♠ ❙●■ ✈❡rs✉s ●■

❙●■

◮ ❉❡s❝❡♥t ◮ ❚❤❡ ♠❛♣ gi → hi ✐s ❛♥ ✐s♦♠♦r♣❤✐s♠ ❜❡t✇❡❡♥ G ❛♥❞ H✦ ◮ ■♥❞❡❡❞✱ ❜② ✶✮ ∃φ : Gn → Hn✳ ◮ ❇② ✷✮ φ t❛❦❡s gn t♦ hn✳ ◮ φ|Gn−1 t❛❦❡s Gn−1 t♦ Hn−1 ❜② ✷✮ ❛♥❞ t❛❦❡s gn−1 t♦ hn−1 ❜② ✸✮✳ ◮ ❚❤❡♥ φ|Gn−1t❛❦❡s Gn−2 t♦ Hn−2 ❛♥❞ ✇❡ ❝❛♥ ❝♦♥t✐♥✉❡ t❤✐s

❞❡s❝❡♥t ♣r♦❝❡ss ✉♥t✐❧ ♦❜t❛✐♥✐♥❣ t❤❛t φ t❛❦❡s ❡✈❡r② gi t♦ hi✳ ❍❡♥❝❡ t❤❡ ♠❛♣ gi → hi ✐s ❛♥ ✐s♦♠♦r♣❤✐s♠ ✐♥ q✉❡st✐♦♥✳

❙❡❛r❝❤✐♥❣ ❢♦r ❛ ●r❛♣❤ ■s♦♠♦r♣❤✐s♠ ❙●■ ✈❡rs✉s ●■

❙●■✿ ❈♦♥str✉❝t✐♥❣ t❤❡ t♦✇❡r

◮ ▲❡t A ❜❡ ❛ ♣♦❧②♥♦♠✐❛❧ ❛❧❣♦r✐t❤♠✱ s♦❧✈✐♥❣ ●■✳

❋✐① ❛♥❞ ❝♦♥s✐❞❡r ❙✐♥❝❡ ✱ s✳t✳ ❋✐♥❞ ❜② ❛♣♣❧②✐♥❣ t♦ ❛❧❧ ♣❛✐rs ✳ ❙❡t ✳ ❛r❡ ✉♥✐q✉❡ ✈❡rt✐❝❡s ♦❢ ❤✐❣❤❡st ❞❡❣r❡❡ ✐♥ ✳ ❍❡♥❝❡ ❡✈❡r② ✐s♦♠♦r♣❤✐s♠ ❜❡t✇❡❡♥ ❛♥❞ t❛❦❡s t♦ ✳ ❊✈❡r② ✐s♦♠♦r♣❤✐s♠ ❜❡t✇❡❡♥ ❛♥❞ t❛❦❡s t♦ ✳ ❚❤✉s ✶✮✲✸✮ ❤♦❧❞ ❢♦r ✳ ❈♦♥t✐♥✉✐♥❣✱ ✜♥❞ t❤❡ ❞❡s✐r❡❞ ❛♥❞ ✱

❙❡❛r❝❤✐♥❣ ❢♦r ❛ ●r❛♣❤ ■s♦♠♦r♣❤✐s♠ ❙●■ ✈❡rs✉s ●■

❙●■✿ ❈♦♥str✉❝t✐♥❣ t❤❡ t♦✇❡r

◮ ▲❡t A ❜❡ ❛ ♣♦❧②♥♦♠✐❛❧ ❛❧❣♦r✐t❤♠✱ s♦❧✈✐♥❣ ●■✳ ◮ ❋✐① k1 ∈ Kn+1 ❛♥❞ ❝♦♥s✐❞❡r

G1 = G ∪g1=k1 Kn+1, H1 (h) = H ∪h=k1 Kn+1. ❙✐♥❝❡ ✱ s✳t✳ ❋✐♥❞ ❜② ❛♣♣❧②✐♥❣ t♦ ❛❧❧ ♣❛✐rs ✳ ❙❡t ✳ ❛r❡ ✉♥✐q✉❡ ✈❡rt✐❝❡s ♦❢ ❤✐❣❤❡st ❞❡❣r❡❡ ✐♥ ✳ ❍❡♥❝❡ ❡✈❡r② ✐s♦♠♦r♣❤✐s♠ ❜❡t✇❡❡♥ ❛♥❞ t❛❦❡s t♦ ✳ ❊✈❡r② ✐s♦♠♦r♣❤✐s♠ ❜❡t✇❡❡♥ ❛♥❞ t❛❦❡s t♦ ✳ ❚❤✉s ✶✮✲✸✮ ❤♦❧❞ ❢♦r ✳ ❈♦♥t✐♥✉✐♥❣✱ ✜♥❞ t❤❡ ❞❡s✐r❡❞ ❛♥❞ ✱

❙❡❛r❝❤✐♥❣ ❢♦r ❛ ●r❛♣❤ ■s♦♠♦r♣❤✐s♠ ❙●■ ✈❡rs✉s ●■

❙●■✿ ❈♦♥str✉❝t✐♥❣ t❤❡ t♦✇❡r

◮ ▲❡t A ❜❡ ❛ ♣♦❧②♥♦♠✐❛❧ ❛❧❣♦r✐t❤♠✱ s♦❧✈✐♥❣ ●■✳ ◮ ❋✐① k1 ∈ Kn+1 ❛♥❞ ❝♦♥s✐❞❡r

G1 = G ∪g1=k1 Kn+1, H1 (h) = H ∪h=k1 Kn+1.

◮ ❙✐♥❝❡ G ≃ H✱

∃h s✳t✳ G1 ≃ H1 = H1 (h) . ❋✐♥❞ ❜② ❛♣♣❧②✐♥❣ t♦ ❛❧❧ ♣❛✐rs ✳ ❙❡t ✳ ❛r❡ ✉♥✐q✉❡ ✈❡rt✐❝❡s ♦❢ ❤✐❣❤❡st ❞❡❣r❡❡ ✐♥ ✳ ❍❡♥❝❡ ❡✈❡r② ✐s♦♠♦r♣❤✐s♠ ❜❡t✇❡❡♥ ❛♥❞ t❛❦❡s t♦ ✳ ❊✈❡r② ✐s♦♠♦r♣❤✐s♠ ❜❡t✇❡❡♥ ❛♥❞ t❛❦❡s t♦ ✳ ❚❤✉s ✶✮✲✸✮ ❤♦❧❞ ❢♦r ✳ ❈♦♥t✐♥✉✐♥❣✱ ✜♥❞ t❤❡ ❞❡s✐r❡❞ ❛♥❞ ✱

❙❡❛r❝❤✐♥❣ ❢♦r ❛ ●r❛♣❤ ■s♦♠♦r♣❤✐s♠ ❙●■ ✈❡rs✉s ●■

❙●■✿ ❈♦♥str✉❝t✐♥❣ t❤❡ t♦✇❡r

◮ ▲❡t A ❜❡ ❛ ♣♦❧②♥♦♠✐❛❧ ❛❧❣♦r✐t❤♠✱ s♦❧✈✐♥❣ ●■✳ ◮ ❋✐① k1 ∈ Kn+1 ❛♥❞ ❝♦♥s✐❞❡r

G1 = G ∪g1=k1 Kn+1, H1 (h) = H ∪h=k1 Kn+1.

◮ ❙✐♥❝❡ G ≃ H✱

∃h s✳t✳ G1 ≃ H1 = H1 (h) .

◮ ❋✐♥❞ h ❜② ❛♣♣❧②✐♥❣ A t♦ ❛❧❧ n ♣❛✐rs (G1, H1(h))✳

❙❡t ✳ ❛r❡ ✉♥✐q✉❡ ✈❡rt✐❝❡s ♦❢ ❤✐❣❤❡st ❞❡❣r❡❡ ✐♥ ✳ ❍❡♥❝❡ ❡✈❡r② ✐s♦♠♦r♣❤✐s♠ ❜❡t✇❡❡♥ ❛♥❞ t❛❦❡s t♦ ✳ ❊✈❡r② ✐s♦♠♦r♣❤✐s♠ ❜❡t✇❡❡♥ ❛♥❞ t❛❦❡s t♦ ✳ ❚❤✉s ✶✮✲✸✮ ❤♦❧❞ ❢♦r ✳ ❈♦♥t✐♥✉✐♥❣✱ ✜♥❞ t❤❡ ❞❡s✐r❡❞ ❛♥❞ ✱

❙❡❛r❝❤✐♥❣ ❢♦r ❛ ●r❛♣❤ ■s♦♠♦r♣❤✐s♠ ❙●■ ✈❡rs✉s ●■

❙●■✿ ❈♦♥str✉❝t✐♥❣ t❤❡ t♦✇❡r

◮ ▲❡t A ❜❡ ❛ ♣♦❧②♥♦♠✐❛❧ ❛❧❣♦r✐t❤♠✱ s♦❧✈✐♥❣ ●■✳ ◮ ❋✐① k1 ∈ Kn+1 ❛♥❞ ❝♦♥s✐❞❡r

G1 = G ∪g1=k1 Kn+1, H1 (h) = H ∪h=k1 Kn+1.

◮ ❙✐♥❝❡ G ≃ H✱

∃h s✳t✳ G1 ≃ H1 = H1 (h) .

◮ ❋✐♥❞ h ❜② ❛♣♣❧②✐♥❣ A t♦ ❛❧❧ n ♣❛✐rs (G1, H1(h))✳ ◮ ❙❡t h1 = h✳

❛r❡ ✉♥✐q✉❡ ✈❡rt✐❝❡s ♦❢ ❤✐❣❤❡st ❞❡❣r❡❡ ✐♥ ✳ ❍❡♥❝❡ ❡✈❡r② ✐s♦♠♦r♣❤✐s♠ ❜❡t✇❡❡♥ ❛♥❞ t❛❦❡s t♦ ✳ ❊✈❡r② ✐s♦♠♦r♣❤✐s♠ ❜❡t✇❡❡♥ ❛♥❞ t❛❦❡s t♦ ✳ ❚❤✉s ✶✮✲✸✮ ❤♦❧❞ ❢♦r ✳ ❈♦♥t✐♥✉✐♥❣✱ ✜♥❞ t❤❡ ❞❡s✐r❡❞ ❛♥❞ ✱

❙❡❛r❝❤✐♥❣ ❢♦r ❛ ●r❛♣❤ ■s♦♠♦r♣❤✐s♠ ❙●■ ✈❡rs✉s ●■

❙●■✿ ❈♦♥str✉❝t✐♥❣ t❤❡ t♦✇❡r

◮ ▲❡t A ❜❡ ❛ ♣♦❧②♥♦♠✐❛❧ ❛❧❣♦r✐t❤♠✱ s♦❧✈✐♥❣ ●■✳ ◮ ❋✐① k1 ∈ Kn+1 ❛♥❞ ❝♦♥s✐❞❡r

G1 = G ∪g1=k1 Kn+1, H1 (h) = H ∪h=k1 Kn+1.

◮ ❙✐♥❝❡ G ≃ H✱

∃h s✳t✳ G1 ≃ H1 = H1 (h) .

◮ ❋✐♥❞ h ❜② ❛♣♣❧②✐♥❣ A t♦ ❛❧❧ n ♣❛✐rs (G1, H1(h))✳ ◮ ❙❡t h1 = h✳ ◮ g1, h1 ❛r❡ ✉♥✐q✉❡ ✈❡rt✐❝❡s ♦❢ ❤✐❣❤❡st ❞❡❣r❡❡ ✐♥ G1, H1✳ ❍❡♥❝❡

❡✈❡r② ✐s♦♠♦r♣❤✐s♠ ❜❡t✇❡❡♥ Gi ❛♥❞ Hi t❛❦❡s gi t♦ hi✳ ❊✈❡r② ✐s♦♠♦r♣❤✐s♠ ❜❡t✇❡❡♥ ❛♥❞ t❛❦❡s t♦ ✳ ❚❤✉s ✶✮✲✸✮ ❤♦❧❞ ❢♦r ✳ ❈♦♥t✐♥✉✐♥❣✱ ✜♥❞ t❤❡ ❞❡s✐r❡❞ ❛♥❞ ✱

❙❡❛r❝❤✐♥❣ ❢♦r ❛ ●r❛♣❤ ■s♦♠♦r♣❤✐s♠ ❙●■ ✈❡rs✉s ●■

❙●■✿ ❈♦♥str✉❝t✐♥❣ t❤❡ t♦✇❡r

◮ ▲❡t A ❜❡ ❛ ♣♦❧②♥♦♠✐❛❧ ❛❧❣♦r✐t❤♠✱ s♦❧✈✐♥❣ ●■✳ ◮ ❋✐① k1 ∈ Kn+1 ❛♥❞ ❝♦♥s✐❞❡r

G1 = G ∪g1=k1 Kn+1, H1 (h) = H ∪h=k1 Kn+1.

◮ ❙✐♥❝❡ G ≃ H✱

∃h s✳t✳ G1 ≃ H1 = H1 (h) .

◮ ❋✐♥❞ h ❜② ❛♣♣❧②✐♥❣ A t♦ ❛❧❧ n ♣❛✐rs (G1, H1(h))✳ ◮ ❙❡t h1 = h✳ ◮ g1, h1 ❛r❡ ✉♥✐q✉❡ ✈❡rt✐❝❡s ♦❢ ❤✐❣❤❡st ❞❡❣r❡❡ ✐♥ G1, H1✳ ❍❡♥❝❡

❡✈❡r② ✐s♦♠♦r♣❤✐s♠ ❜❡t✇❡❡♥ Gi ❛♥❞ Hi t❛❦❡s gi t♦ hi✳

◮ ❊✈❡r② ✐s♦♠♦r♣❤✐s♠ ❜❡t✇❡❡♥ G1 ❛♥❞ H1 t❛❦❡s G0 t♦ H0✳

❚❤✉s ✶✮✲✸✮ ❤♦❧❞ ❢♦r i = 1✳ ❈♦♥t✐♥✉✐♥❣✱ ✜♥❞ t❤❡ ❞❡s✐r❡❞ ❛♥❞ ✱

❙❡❛r❝❤✐♥❣ ❢♦r ❛ ●r❛♣❤ ■s♦♠♦r♣❤✐s♠ ❙●■ ✈❡rs✉s ●■

❙●■✿ ❈♦♥str✉❝t✐♥❣ t❤❡ t♦✇❡r

◮ ▲❡t A ❜❡ ❛ ♣♦❧②♥♦♠✐❛❧ ❛❧❣♦r✐t❤♠✱ s♦❧✈✐♥❣ ●■✳ ◮ ❋✐① k1 ∈ Kn+1 ❛♥❞ ❝♦♥s✐❞❡r

G1 = G ∪g1=k1 Kn+1, H1 (h) = H ∪h=k1 Kn+1.

◮ ❙✐♥❝❡ G ≃ H✱

∃h s✳t✳ G1 ≃ H1 = H1 (h) .

◮ ❋✐♥❞ h ❜② ❛♣♣❧②✐♥❣ A t♦ ❛❧❧ n ♣❛✐rs (G1, H1(h))✳ ◮ ❙❡t h1 = h✳ ◮ g1, h1 ❛r❡ ✉♥✐q✉❡ ✈❡rt✐❝❡s ♦❢ ❤✐❣❤❡st ❞❡❣r❡❡ ✐♥ G1, H1✳ ❍❡♥❝❡

❡✈❡r② ✐s♦♠♦r♣❤✐s♠ ❜❡t✇❡❡♥ Gi ❛♥❞ Hi t❛❦❡s gi t♦ hi✳

◮ ❊✈❡r② ✐s♦♠♦r♣❤✐s♠ ❜❡t✇❡❡♥ G1 ❛♥❞ H1 t❛❦❡s G0 t♦ H0✳

❚❤✉s ✶✮✲✸✮ ❤♦❧❞ ❢♦r i = 1✳

◮ ❈♦♥t✐♥✉✐♥❣✱ ✜♥❞ t❤❡ ❞❡s✐r❡❞ h1, . . . , hn ∈ H

❛♥❞ G = G0 < G1 < · · · < Gn✱ H = H0 < H1 < · · · < Hn.

❙❡❛r❝❤✐♥❣ ❢♦r ❛ ●r❛♣❤ ■s♦♠♦r♣❤✐s♠ ❙●■ ✈❡rs✉s ●■

❙●■✿ ❈♦♥str✉❝t✐♥❣ t❤❡ t♦✇❡r

◮ ▲❡t A ❜❡ ❛ ♣♦❧②♥♦♠✐❛❧ ❛❧❣♦r✐t❤♠✱ s♦❧✈✐♥❣ ●■✳ ◮ ❋✐① k1 ∈ Kn+1 ❛♥❞ ❝♦♥s✐❞❡r

G1 = G ∪g1=k1 Kn+1, H1 (h) = H ∪h=k1 Kn+1.

◮ ❙✐♥❝❡ G ≃ H✱

∃h s✳t✳ G1 ≃ H1 = H1 (h) .

◮ ❋✐♥❞ h ❜② ❛♣♣❧②✐♥❣ A t♦ ❛❧❧ n ♣❛✐rs (G1, H1(h))✳ ◮ ❙❡t h1 = h✳ ◮ g1, h1 ❛r❡ ✉♥✐q✉❡ ✈❡rt✐❝❡s ♦❢ ❤✐❣❤❡st ❞❡❣r❡❡ ✐♥ G1, H1✳ ❍❡♥❝❡

❡✈❡r② ✐s♦♠♦r♣❤✐s♠ ❜❡t✇❡❡♥ Gi ❛♥❞ Hi t❛❦❡s gi t♦ hi✳

◮ ❊✈❡r② ✐s♦♠♦r♣❤✐s♠ ❜❡t✇❡❡♥ G1 ❛♥❞ H1 t❛❦❡s G0 t♦ H0✳

❚❤✉s ✶✮✲✸✮ ❤♦❧❞ ❢♦r i = 1✳

◮ ❈♦♥t✐♥✉✐♥❣✱ ✜♥❞ t❤❡ ❞❡s✐r❡❞ h1, . . . , hn ∈ H

❛♥❞ G = G0 < G1 < · · · < Gn✱ H = H0 < H1 < · · · < Hn.

❙❡❛r❝❤✐♥❣ ❢♦r ❛ ●r❛♣❤ ■s♦♠♦r♣❤✐s♠ ❆❧❣♦r✐t❤♠ ❢♦r ❙●■

Pr♦❜❛❜✐❧✐s❧t✐❝ ❛❧❣♦r✐t❤♠ ❢♦r ❙●■

◮ ❚❤❡♦r❡♠ ✭❆✳◆✳❘②❜❛❧♦✈✱ ✷✵✶✺✮

■❢ t❤❡r❡ ❡①✐sts ❛ ♣♦❧②♥♦♠✐❛❧ ❣❡♥❡r✐❝ ❛❧❣♦r✐t❤♠ ❢♦r ❙●■✱ t❤❡♥ t❤❡r❡ ❡①✐sts ❛ ♣♦❧②♥♦♠✐❛❧ ♣r♦❜❛❜✐❧✐st✐❝ ❛❧❣♦r✐t❤♠ ❝♦♠♣✉t✐♥❣ ❙●■ ❢♦r ❛❧❧ ✐♥♣✉ts✳

❚❤❡♦r❡♠ ✭●✳❆✳◆♦s❦♦✈✱✷✵✶✺✮

❚❤❡r❡ ✐s ❛ ♣♦❧②♥♦♠✐❛❧ ❛❧❣♦r✐t❤♠ s♦❧✈✐♥❣ ❙●■ ♦♥ ❛♥ ❛s②♠♣t♦t✐❝❛❧❧② ❛❧♠♦st ❛❧❧ ✐♥♣✉ts✳

❙❡❛r❝❤✐♥❣ ❢♦r ❛ ●r❛♣❤ ■s♦♠♦r♣❤✐s♠ ❆❧❣♦r✐t❤♠ ❢♦r ❙●■

Pr♦❜❛❜✐❧✐s❧t✐❝ ❛❧❣♦r✐t❤♠ ❢♦r ❙●■

◮ ❚❤❡♦r❡♠ ✭❆✳◆✳❘②❜❛❧♦✈✱ ✷✵✶✺✮

■❢ t❤❡r❡ ❡①✐sts ❛ ♣♦❧②♥♦♠✐❛❧ ❣❡♥❡r✐❝ ❛❧❣♦r✐t❤♠ ❢♦r ❙●■✱ t❤❡♥ t❤❡r❡ ❡①✐sts ❛ ♣♦❧②♥♦♠✐❛❧ ♣r♦❜❛❜✐❧✐st✐❝ ❛❧❣♦r✐t❤♠ ❝♦♠♣✉t✐♥❣ ❙●■ ❢♦r ❛❧❧ ✐♥♣✉ts✳

◮ ❚❤❡♦r❡♠ ✭●✳❆✳◆♦s❦♦✈✱✷✵✶✺✮

❚❤❡r❡ ✐s ❛ ♣♦❧②♥♦♠✐❛❧ ❛❧❣♦r✐t❤♠ s♦❧✈✐♥❣ ❙●■ ♦♥ ❛♥ ❛s②♠♣t♦t✐❝❛❧❧② ❛❧♠♦st ❛❧❧ ✐♥♣✉ts✳

❙❡❛r❝❤✐♥❣ ❢♦r ❛ ●r❛♣❤ ■s♦♠♦r♣❤✐s♠ ❆❧❣♦r✐t❤♠ ❢♦r ❙●■

Pr♦❜❛❜✐❧✐s❧t✐❝ ❛❧❣♦r✐t❤♠ ❢♦r ❙●■

◮ ❚❤❡♦r❡♠ ✭❆✳◆✳❘②❜❛❧♦✈✱ ✷✵✶✺✮

■❢ t❤❡r❡ ❡①✐sts ❛ ♣♦❧②♥♦♠✐❛❧ ❣❡♥❡r✐❝ ❛❧❣♦r✐t❤♠ ❢♦r ❙●■✱ t❤❡♥ t❤❡r❡ ❡①✐sts ❛ ♣♦❧②♥♦♠✐❛❧ ♣r♦❜❛❜✐❧✐st✐❝ ❛❧❣♦r✐t❤♠ ❝♦♠♣✉t✐♥❣ ❙●■ ❢♦r ❛❧❧ ✐♥♣✉ts✳

◮ ❚❤❡♦r❡♠ ✭●✳❆✳◆♦s❦♦✈✱✷✵✶✺✮

❚❤❡r❡ ✐s ❛ ♣♦❧②♥♦♠✐❛❧ ❛❧❣♦r✐t❤♠ s♦❧✈✐♥❣ ❙●■ ♦♥ ❛♥ ❛s②♠♣t♦t✐❝❛❧❧② ❛❧♠♦st ❛❧❧ ✐♥♣✉ts✳

❙❡❛r❝❤✐♥❣ ❢♦r ❛ ●r❛♣❤ ■s♦♠♦r♣❤✐s♠ ❆❧❣♦r✐t❤♠ ❢♦r ❙●■

❙●■✿●r❛♣❤ ✐♥✈❛r✐❛♥t τ

◮ ❚❤❡ t②♣❡ τ(v) ∈ N∗ ♦❢ ❛ ✈❡rt❡① v ✐s t❤❡ str✐♥❣ ♦❢ ❞❡❣r❡❡s ♦❢

✈❡rt✐❝❡s ♦❢ N (v) ✐♥ ♥♦♥✲❞❡❝r❡❛s✐♥❣ ♦r❞❡r✳ ■♥ ❞❡t❛✐❧✱ ✱ ✇❤❡r❡ ✳ ❚❤❡ t②♣❡ ✐s ♣r❡s❡r✈❡❞ ✉♥❞❡r ❣r❛♣❤ ✐s♦♠♦r♣❤✐s♠s✿ ✐❢ ✐s ❛ ❣r❛♣❤ ✐s♦♠♦r♣❤✐s♠ t❤❡♥ ❢♦r ❛❧❧ ✱ ✐✳❡✳ ✳ ❚❤❡ t②♣❡ ♦❢ ✐s t❤❡ ▲❡① ♦r❞❡r❡❞ s❡q✉❡♥❝❡ ♦❢ ❛❧❧ t②♣❡s ♦❢ ❛❧❧ ✈❡rt✐❝❡s✳ ❚❤❡ t②♣❡ ❢✉♥❝t✐♦♥ ✐s ❛ ❣r❛♣❤ ✐♥✈❛r✐❛♥t ✱ ✐✳❡✳ ❢♦r ✐s♦♠♦r♣❤✐❝ ❣r❛♣❤s ✳

❙❡❛r❝❤✐♥❣ ❢♦r ❛ ●r❛♣❤ ■s♦♠♦r♣❤✐s♠ ❆❧❣♦r✐t❤♠ ❢♦r ❙●■

❙●■✿●r❛♣❤ ✐♥✈❛r✐❛♥t τ

◮ ❚❤❡ t②♣❡ τ(v) ∈ N∗ ♦❢ ❛ ✈❡rt❡① v ✐s t❤❡ str✐♥❣ ♦❢ ❞❡❣r❡❡s ♦❢

✈❡rt✐❝❡s ♦❢ N (v) ✐♥ ♥♦♥✲❞❡❝r❡❛s✐♥❣ ♦r❞❡r✳

◮ ■♥ ❞❡t❛✐❧✱ τ (v) = (d1, ..., dd(v))✱ ✇❤❡r❡ d1 ≤ d2 ≤ . . . ≤ dd(v)✳

❚❤❡ t②♣❡ ✐s ♣r❡s❡r✈❡❞ ✉♥❞❡r ❣r❛♣❤ ✐s♦♠♦r♣❤✐s♠s✿ ✐❢ ✐s ❛ ❣r❛♣❤ ✐s♦♠♦r♣❤✐s♠ t❤❡♥ ❢♦r ❛❧❧ ✱ ✐✳❡✳ ✳ ❚❤❡ t②♣❡ ♦❢ ✐s t❤❡ ▲❡① ♦r❞❡r❡❞ s❡q✉❡♥❝❡ ♦❢ ❛❧❧ t②♣❡s ♦❢ ❛❧❧ ✈❡rt✐❝❡s✳ ❚❤❡ t②♣❡ ❢✉♥❝t✐♦♥ ✐s ❛ ❣r❛♣❤ ✐♥✈❛r✐❛♥t ✱ ✐✳❡✳ ❢♦r ✐s♦♠♦r♣❤✐❝ ❣r❛♣❤s ✳

❙❡❛r❝❤✐♥❣ ❢♦r ❛ ●r❛♣❤ ■s♦♠♦r♣❤✐s♠ ❆❧❣♦r✐t❤♠ ❢♦r ❙●■

❙●■✿●r❛♣❤ ✐♥✈❛r✐❛♥t τ

◮ ❚❤❡ t②♣❡ τ(v) ∈ N∗ ♦❢ ❛ ✈❡rt❡① v ✐s t❤❡ str✐♥❣ ♦❢ ❞❡❣r❡❡s ♦❢

✈❡rt✐❝❡s ♦❢ N (v) ✐♥ ♥♦♥✲❞❡❝r❡❛s✐♥❣ ♦r❞❡r✳

◮ ■♥ ❞❡t❛✐❧✱ τ (v) = (d1, ..., dd(v))✱ ✇❤❡r❡ d1 ≤ d2 ≤ . . . ≤ dd(v)✳ ◮ ❚❤❡ t②♣❡ ✐s ♣r❡s❡r✈❡❞ ✉♥❞❡r ❣r❛♣❤ ✐s♦♠♦r♣❤✐s♠s✿ ✐❢ φ : G → H

✐s ❛ ❣r❛♣❤ ✐s♦♠♦r♣❤✐s♠ t❤❡♥ τ (φ (v)) = τ (v) ❢♦r ❛❧❧ v ∈ V (G)✱ ✐✳❡✳ τ ◦ φ = τ✳ ❚❤❡ t②♣❡ ♦❢ ✐s t❤❡ ▲❡① ♦r❞❡r❡❞ s❡q✉❡♥❝❡ ♦❢ ❛❧❧ t②♣❡s ♦❢ ❛❧❧ ✈❡rt✐❝❡s✳ ❚❤❡ t②♣❡ ❢✉♥❝t✐♦♥ ✐s ❛ ❣r❛♣❤ ✐♥✈❛r✐❛♥t ✱ ✐✳❡✳ ❢♦r ✐s♦♠♦r♣❤✐❝ ❣r❛♣❤s ✳

❙❡❛r❝❤✐♥❣ ❢♦r ❛ ●r❛♣❤ ■s♦♠♦r♣❤✐s♠ ❆❧❣♦r✐t❤♠ ❢♦r ❙●■

❙●■✿●r❛♣❤ ✐♥✈❛r✐❛♥t τ

◮ ❚❤❡ t②♣❡ τ(v) ∈ N∗ ♦❢ ❛ ✈❡rt❡① v ✐s t❤❡ str✐♥❣ ♦❢ ❞❡❣r❡❡s ♦❢

✈❡rt✐❝❡s ♦❢ N (v) ✐♥ ♥♦♥✲❞❡❝r❡❛s✐♥❣ ♦r❞❡r✳

◮ ■♥ ❞❡t❛✐❧✱ τ (v) = (d1, ..., dd(v))✱ ✇❤❡r❡ d1 ≤ d2 ≤ . . . ≤ dd(v)✳ ◮ ❚❤❡ t②♣❡ ✐s ♣r❡s❡r✈❡❞ ✉♥❞❡r ❣r❛♣❤ ✐s♦♠♦r♣❤✐s♠s✿ ✐❢ φ : G → H

✐s ❛ ❣r❛♣❤ ✐s♦♠♦r♣❤✐s♠ t❤❡♥ τ (φ (v)) = τ (v) ❢♦r ❛❧❧ v ∈ V (G)✱ ✐✳❡✳ τ ◦ φ = τ✳

◮ ❚❤❡ t②♣❡ τG ♦❢ G ✐s t❤❡ ▲❡① ♦r❞❡r❡❞ s❡q✉❡♥❝❡

(τ (v1) , . . . , τ (vn)) , n = |V | ♦❢ ❛❧❧ t②♣❡s ♦❢ ❛❧❧ ✈❡rt✐❝❡s✳ ❚❤❡ t②♣❡ ❢✉♥❝t✐♦♥ ✐s ❛ ❣r❛♣❤ ✐♥✈❛r✐❛♥t ✱ ✐✳❡✳ ❢♦r ✐s♦♠♦r♣❤✐❝ ❣r❛♣❤s ✳

❙❡❛r❝❤✐♥❣ ❢♦r ❛ ●r❛♣❤ ■s♦♠♦r♣❤✐s♠ ❆❧❣♦r✐t❤♠ ❢♦r ❙●■

❙●■✿●r❛♣❤ ✐♥✈❛r✐❛♥t τ

◮ ❚❤❡ t②♣❡ τ(v) ∈ N∗ ♦❢ ❛ ✈❡rt❡① v ✐s t❤❡ str✐♥❣ ♦❢ ❞❡❣r❡❡s ♦❢

✈❡rt✐❝❡s ♦❢ N (v) ✐♥ ♥♦♥✲❞❡❝r❡❛s✐♥❣ ♦r❞❡r✳

◮ ■♥ ❞❡t❛✐❧✱ τ (v) = (d1, ..., dd(v))✱ ✇❤❡r❡ d1 ≤ d2 ≤ . . . ≤ dd(v)✳ ◮ ❚❤❡ t②♣❡ ✐s ♣r❡s❡r✈❡❞ ✉♥❞❡r ❣r❛♣❤ ✐s♦♠♦r♣❤✐s♠s✿ ✐❢ φ : G → H

✐s ❛ ❣r❛♣❤ ✐s♦♠♦r♣❤✐s♠ t❤❡♥ τ (φ (v)) = τ (v) ❢♦r ❛❧❧ v ∈ V (G)✱ ✐✳❡✳ τ ◦ φ = τ✳

◮ ❚❤❡ t②♣❡ τG ♦❢ G ✐s t❤❡ ▲❡① ♦r❞❡r❡❞ s❡q✉❡♥❝❡

(τ (v1) , . . . , τ (vn)) , n = |V | ♦❢ ❛❧❧ t②♣❡s ♦❢ ❛❧❧ ✈❡rt✐❝❡s✳

◮ ❚❤❡ t②♣❡ ❢✉♥❝t✐♦♥ G → τG ✐s ❛ ❣r❛♣❤ ✐♥✈❛r✐❛♥t ✱ ✐✳❡✳ τG = τH

❢♦r ✐s♦♠♦r♣❤✐❝ ❣r❛♣❤s G, H✳

❙❡❛r❝❤✐♥❣ ❢♦r ❛ ●r❛♣❤ ■s♦♠♦r♣❤✐s♠ ❆❧❣♦r✐t❤♠ ❢♦r ❙●■

❙●■✿●r❛♣❤ ✐♥✈❛r✐❛♥t τ

◮ ❚❤❡ t②♣❡ τ(v) ∈ N∗ ♦❢ ❛ ✈❡rt❡① v ✐s t❤❡ str✐♥❣ ♦❢ ❞❡❣r❡❡s ♦❢

✈❡rt✐❝❡s ♦❢ N (v) ✐♥ ♥♦♥✲❞❡❝r❡❛s✐♥❣ ♦r❞❡r✳

◮ ■♥ ❞❡t❛✐❧✱ τ (v) = (d1, ..., dd(v))✱ ✇❤❡r❡ d1 ≤ d2 ≤ . . . ≤ dd(v)✳ ◮ ❚❤❡ t②♣❡ ✐s ♣r❡s❡r✈❡❞ ✉♥❞❡r ❣r❛♣❤ ✐s♦♠♦r♣❤✐s♠s✿ ✐❢ φ : G → H

✐s ❛ ❣r❛♣❤ ✐s♦♠♦r♣❤✐s♠ t❤❡♥ τ (φ (v)) = τ (v) ❢♦r ❛❧❧ v ∈ V (G)✱ ✐✳❡✳ τ ◦ φ = τ✳

◮ ❚❤❡ t②♣❡ τG ♦❢ G ✐s t❤❡ ▲❡① ♦r❞❡r❡❞ s❡q✉❡♥❝❡

(τ (v1) , . . . , τ (vn)) , n = |V | ♦❢ ❛❧❧ t②♣❡s ♦❢ ❛❧❧ ✈❡rt✐❝❡s✳

◮ ❚❤❡ t②♣❡ ❢✉♥❝t✐♦♥ G → τG ✐s ❛ ❣r❛♣❤ ✐♥✈❛r✐❛♥t ✱ ✐✳❡✳ τG = τH

❢♦r ✐s♦♠♦r♣❤✐❝ ❣r❛♣❤s G, H✳

❙❡❛r❝❤✐♥❣ ❢♦r ❛ ●r❛♣❤ ■s♦♠♦r♣❤✐s♠ ❆❧❣♦r✐t❤♠ ❢♦r ❙●■

❖❜❧✐q✉❡ ❣r❛♣❤s

◮ ❏✳ ❙❝❤r❡②❡r✱ ❍✳❲❛❧t❤❡r✱ ▲✳❙✳ ▼❡❧♥✐❦♦✈✱ ❱❡rt❡①✲♦❜❧✐q✉❡ ❣r❛♣❤s✱

❉✐s❝r❡t❡ ▼❛t❤✳ ✸✵✼ ✭✷✵✵✼✮✳ ❆ ❣r❛♣❤ ✐s s❛✐❞ t♦ ❜❡ ♦❜❧✐q✉❡ ✐❢ t❤❡ ♠❛♣ ✐s ✐♥❥❡❝t✐✈❡ ♦r✱ ✐♥ ♦t❤❡r ✇♦r❞s✱ t❤❡r❡ ❛r❡ ♥♦ ❞✐st✐♥❝t ✈❡rt✐❝❡s s✉❝❤ t❤❛t ✳ ♣❛✉s❡ ■❢ ❛ ❣r❛♣❤ ✐s ♦❜❧✐q✉❡ t❤❡♥ ✐ts t②♣❡ ✈❡❝t♦r ✐s ✉♥✐q✉❡ ✐♥ t❤❡ s❡♥s❡ t❤❛t ✐♠♣❧✐❡s t❤❛t ❢♦r ❛❧❧ ✳

▲❡♠♠❛ ✭r❡tr✐❡✈✐♥❣ ❛♥ ✐s♦♠♦r♣❤✐s♠✮

▲❡t ❜❡ t❤❡ t②♣❡✲✈❡❝t♦rs ♦❢ r❡s♣❡❝t✐✈❡❧②✳ ■❢ ✐s ♦❜❧✐q✉❡ ❛♥❞ t❤❡♥ t❤❡ ♠❛♣ ✐s t❤❡ ✐s♦♠♦r♣❤✐s♠ ♦❢ ♦♥t♦ ❛♥❞ t❤❡r❡ ✐s ♥♦ ♦t❤❡r ✐s♦♠♦r♣❤✐s♠ ❜❡t✇❡❡♥ ❛♥❞ ✳ ■♥ ♣❛rt✐❝✉❧❛r✱ ❡✈❡r② ♦❜❧✐q✉❡ ❣r❛♣❤ ❤❛s ❛ tr✐✈✐❛❧ ❛✉t♦♠♦r♣❤✐s♠ ❣r♦✉♣✳

❙❡❛r❝❤✐♥❣ ❢♦r ❛ ●r❛♣❤ ■s♦♠♦r♣❤✐s♠ ❆❧❣♦r✐t❤♠ ❢♦r ❙●■

❖❜❧✐q✉❡ ❣r❛♣❤s

◮ ❏✳ ❙❝❤r❡②❡r✱ ❍✳❲❛❧t❤❡r✱ ▲✳❙✳ ▼❡❧♥✐❦♦✈✱ ❱❡rt❡①✲♦❜❧✐q✉❡ ❣r❛♣❤s✱

❉✐s❝r❡t❡ ▼❛t❤✳ ✸✵✼ ✭✷✵✵✼✮✳

◮ ❆ ❣r❛♣❤ G ✐s s❛✐❞ t♦ ❜❡ ♦❜❧✐q✉❡ ✐❢ t❤❡ ♠❛♣ τ : V → N∗ ✐s

✐♥❥❡❝t✐✈❡ ♦r✱ ✐♥ ♦t❤❡r ✇♦r❞s✱ t❤❡r❡ ❛r❡ ♥♦ ❞✐st✐♥❝t ✈❡rt✐❝❡s u, v ∈ V (G) s✉❝❤ t❤❛t τ (v) = τ (w)✳ ♣❛✉s❡

◮ ■❢ ❛ ❣r❛♣❤ G ✐s ♦❜❧✐q✉❡ t❤❡♥ ✐ts t②♣❡ ✈❡❝t♦r

τG = (τ (v1) , . . . , τ (vn)) ✐s ✉♥✐q✉❡ ✐♥ t❤❡ s❡♥s❡ t❤❛t (τ (v1) , . . . , τ (vn)) = (τ (w1) , . . . , τ (wn)) ✐♠♣❧✐❡s t❤❛t vi = wi ❢♦r ❛❧❧ i✳

▲❡♠♠❛ ✭r❡tr✐❡✈✐♥❣ ❛♥ ✐s♦♠♦r♣❤✐s♠✮

▲❡t ❜❡ t❤❡ t②♣❡✲✈❡❝t♦rs ♦❢ r❡s♣❡❝t✐✈❡❧②✳ ■❢ ✐s ♦❜❧✐q✉❡ ❛♥❞ t❤❡♥ t❤❡ ♠❛♣ ✐s t❤❡ ✐s♦♠♦r♣❤✐s♠ ♦❢ ♦♥t♦ ❛♥❞ t❤❡r❡ ✐s ♥♦ ♦t❤❡r ✐s♦♠♦r♣❤✐s♠ ❜❡t✇❡❡♥ ❛♥❞ ✳ ■♥ ♣❛rt✐❝✉❧❛r✱ ❡✈❡r② ♦❜❧✐q✉❡ ❣r❛♣❤ ❤❛s ❛ tr✐✈✐❛❧ ❛✉t♦♠♦r♣❤✐s♠ ❣r♦✉♣✳

❙❡❛r❝❤✐♥❣ ❢♦r ❛ ●r❛♣❤ ■s♦♠♦r♣❤✐s♠ ❆❧❣♦r✐t❤♠ ❢♦r ❙●■

❖❜❧✐q✉❡ ❣r❛♣❤s

◮ ❏✳ ❙❝❤r❡②❡r✱ ❍✳❲❛❧t❤❡r✱ ▲✳❙✳ ▼❡❧♥✐❦♦✈✱ ❱❡rt❡①✲♦❜❧✐q✉❡ ❣r❛♣❤s✱

❉✐s❝r❡t❡ ▼❛t❤✳ ✸✵✼ ✭✷✵✵✼✮✳

◮ ❆ ❣r❛♣❤ G ✐s s❛✐❞ t♦ ❜❡ ♦❜❧✐q✉❡ ✐❢ t❤❡ ♠❛♣ τ : V → N∗ ✐s

✐♥❥❡❝t✐✈❡ ♦r✱ ✐♥ ♦t❤❡r ✇♦r❞s✱ t❤❡r❡ ❛r❡ ♥♦ ❞✐st✐♥❝t ✈❡rt✐❝❡s u, v ∈ V (G) s✉❝❤ t❤❛t τ (v) = τ (w)✳ ♣❛✉s❡

◮ ■❢ ❛ ❣r❛♣❤ G ✐s ♦❜❧✐q✉❡ t❤❡♥ ✐ts t②♣❡ ✈❡❝t♦r

τG = (τ (v1) , . . . , τ (vn)) ✐s ✉♥✐q✉❡ ✐♥ t❤❡ s❡♥s❡ t❤❛t (τ (v1) , . . . , τ (vn)) = (τ (w1) , . . . , τ (wn)) ✐♠♣❧✐❡s t❤❛t vi = wi ❢♦r ❛❧❧ i✳

◮ ▲❡♠♠❛ ✭r❡tr✐❡✈✐♥❣ ❛♥ ✐s♦♠♦r♣❤✐s♠✮

▲❡t τG = (τ (g1) , . . . , τ (gn)) , τH = (τ (h1) , . . . , τ (hn)) ❜❡ t❤❡ t②♣❡✲✈❡❝t♦rs ♦❢ G, H r❡s♣❡❝t✐✈❡❧②✳ ■❢ G ✐s ♦❜❧✐q✉❡ ❛♥❞ G ≃ H t❤❡♥ t❤❡ ♠❛♣ gi → hi ✐s t❤❡ ✐s♦♠♦r♣❤✐s♠ ♦❢ G ♦♥t♦ H ❛♥❞ t❤❡r❡ ✐s ♥♦ ♦t❤❡r ✐s♦♠♦r♣❤✐s♠ ❜❡t✇❡❡♥ G ❛♥❞ H✳ ■♥ ♣❛rt✐❝✉❧❛r✱ ❡✈❡r② ♦❜❧✐q✉❡ ❣r❛♣❤ ❤❛s ❛ tr✐✈✐❛❧ ❛✉t♦♠♦r♣❤✐s♠ ❣r♦✉♣✳

❙❡❛r❝❤✐♥❣ ❢♦r ❛ ●r❛♣❤ ■s♦♠♦r♣❤✐s♠ ❆❧❣♦r✐t❤♠ ❢♦r ❙●■

❖❜❧✐q✉❡ ❣r❛♣❤s

◮ ❏✳ ❙❝❤r❡②❡r✱ ❍✳❲❛❧t❤❡r✱ ▲✳❙✳ ▼❡❧♥✐❦♦✈✱ ❱❡rt❡①✲♦❜❧✐q✉❡ ❣r❛♣❤s✱

❉✐s❝r❡t❡ ▼❛t❤✳ ✸✵✼ ✭✷✵✵✼✮✳

◮ ❆ ❣r❛♣❤ G ✐s s❛✐❞ t♦ ❜❡ ♦❜❧✐q✉❡ ✐❢ t❤❡ ♠❛♣ τ : V → N∗ ✐s

✐♥❥❡❝t✐✈❡ ♦r✱ ✐♥ ♦t❤❡r ✇♦r❞s✱ t❤❡r❡ ❛r❡ ♥♦ ❞✐st✐♥❝t ✈❡rt✐❝❡s u, v ∈ V (G) s✉❝❤ t❤❛t τ (v) = τ (w)✳ ♣❛✉s❡

◮ ■❢ ❛ ❣r❛♣❤ G ✐s ♦❜❧✐q✉❡ t❤❡♥ ✐ts t②♣❡ ✈❡❝t♦r

τG = (τ (v1) , . . . , τ (vn)) ✐s ✉♥✐q✉❡ ✐♥ t❤❡ s❡♥s❡ t❤❛t (τ (v1) , . . . , τ (vn)) = (τ (w1) , . . . , τ (wn)) ✐♠♣❧✐❡s t❤❛t vi = wi ❢♦r ❛❧❧ i✳

◮ ▲❡♠♠❛ ✭r❡tr✐❡✈✐♥❣ ❛♥ ✐s♦♠♦r♣❤✐s♠✮

▲❡t τG = (τ (g1) , . . . , τ (gn)) , τH = (τ (h1) , . . . , τ (hn)) ❜❡ t❤❡ t②♣❡✲✈❡❝t♦rs ♦❢ G, H r❡s♣❡❝t✐✈❡❧②✳ ■❢ G ✐s ♦❜❧✐q✉❡ ❛♥❞ G ≃ H t❤❡♥ t❤❡ ♠❛♣ gi → hi ✐s t❤❡ ✐s♦♠♦r♣❤✐s♠ ♦❢ G ♦♥t♦ H ❛♥❞ t❤❡r❡ ✐s ♥♦ ♦t❤❡r ✐s♦♠♦r♣❤✐s♠ ❜❡t✇❡❡♥ G ❛♥❞ H✳ ■♥ ♣❛rt✐❝✉❧❛r✱ ❡✈❡r② ♦❜❧✐q✉❡ ❣r❛♣❤ ❤❛s ❛ tr✐✈✐❛❧ ❛✉t♦♠♦r♣❤✐s♠ ❣r♦✉♣✳

❙❡❛r❝❤✐♥❣ ❢♦r ❛ ●r❛♣❤ ■s♦♠♦r♣❤✐s♠ ❆❧❣♦r✐t❤♠ ❢♦r ❙●■

❙●■✿ P❛rt✐❛❧ ♣♦❧②♥♦♠✐❛❧ ❛❧❣♦r✐t❤♠ A

◮ ■♥♣✉t s❡t✿ I =∪ In✱ ✇❤❡r❡

In = {(G, H) : G, H ∈ Gn, G ≃ H}✳ ❚❤❡ ♣r♦❜❧❡♠✿ ❣✐✈❡♥ t♦ ❝♦♥str✉❝t s✉❝❤ t❤❛t ✳

• ✐✈❡♥ ✐s♦♠♦r♣❤✐❝ ❣r❛♣❤s

✱ t❤❡ ❛❧❣♦r✐t❤♠ ✜rst ❝♦♠♣✉t❡s t❤❡✐r t②♣❡✲✈❡❝t♦rs ■❢ t❤❡ ❝♦♦r❞✐♥❛t❡s ♦❢ ❛t ❧❡❛st ♦♥❡ ✈❡❝t♦r ❛r❡ ♥♦t ❞✐st✐♥❝t✱ t❤❡ ❛❧❣♦r✐t❤♠ ❛♥s✇❡rs ✧❄✧✭❞♦♥✬t ❦♥♦✇✮✳ ❖t❤❡r✇✐s❡ ❛r❡ ❜♦t❤ ♦❜❧✐q✉❡ ❛♥❞ ❝❤❡❝❦s ✇❤❡t❤❡r t❤❡ ♠❛♣ ✐s ❛♥ ✐s♦♠♦r♣❤✐s♠ ♦r ♥♦t✳ ■❢ t❤✐s ♠❛♣ t✉r♥s ♦✉t t♦ ❜❡ ❛♥ ✐s♦♠♦r♣❤✐s♠✱ t❤❡♥ t❤❡ ❛❧❣♦r✐t❤♠ ❣✐✈❡s ❛s t❤❡ ❛♥s✇❡r t♦ t❤❡ ♣r♦❜❧❡♠✳

❙❡❛r❝❤✐♥❣ ❢♦r ❛ ●r❛♣❤ ■s♦♠♦r♣❤✐s♠ ❆❧❣♦r✐t❤♠ ❢♦r ❙●■

❙●■✿ P❛rt✐❛❧ ♣♦❧②♥♦♠✐❛❧ ❛❧❣♦r✐t❤♠ A

◮ ■♥♣✉t s❡t✿ I =∪ In✱ ✇❤❡r❡

In = {(G, H) : G, H ∈ Gn, G ≃ H}✳

◮ ❚❤❡ ♣r♦❜❧❡♠✿ ❣✐✈❡♥ (G, H) ∈ In t♦ ❝♦♥str✉❝t π ∈ Sn s✉❝❤

t❤❛t πG = H✳

• ✐✈❡♥ ✐s♦♠♦r♣❤✐❝ ❣r❛♣❤s

✱ t❤❡ ❛❧❣♦r✐t❤♠ ✜rst ❝♦♠♣✉t❡s t❤❡✐r t②♣❡✲✈❡❝t♦rs ■❢ t❤❡ ❝♦♦r❞✐♥❛t❡s ♦❢ ❛t ❧❡❛st ♦♥❡ ✈❡❝t♦r ❛r❡ ♥♦t ❞✐st✐♥❝t✱ t❤❡ ❛❧❣♦r✐t❤♠ ❛♥s✇❡rs ✧❄✧✭❞♦♥✬t ❦♥♦✇✮✳ ❖t❤❡r✇✐s❡ ❛r❡ ❜♦t❤ ♦❜❧✐q✉❡ ❛♥❞ ❝❤❡❝❦s ✇❤❡t❤❡r t❤❡ ♠❛♣ ✐s ❛♥ ✐s♦♠♦r♣❤✐s♠ ♦r ♥♦t✳ ■❢ t❤✐s ♠❛♣ t✉r♥s ♦✉t t♦ ❜❡ ❛♥ ✐s♦♠♦r♣❤✐s♠✱ t❤❡♥ t❤❡ ❛❧❣♦r✐t❤♠ ❣✐✈❡s ❛s t❤❡ ❛♥s✇❡r t♦ t❤❡ ♣r♦❜❧❡♠✳

❙❡❛r❝❤✐♥❣ ❢♦r ❛ ●r❛♣❤ ■s♦♠♦r♣❤✐s♠ ❆❧❣♦r✐t❤♠ ❢♦r ❙●■

❙●■✿ P❛rt✐❛❧ ♣♦❧②♥♦♠✐❛❧ ❛❧❣♦r✐t❤♠ A

◮ ■♥♣✉t s❡t✿ I =∪ In✱ ✇❤❡r❡

In = {(G, H) : G, H ∈ Gn, G ≃ H}✳

◮ ❚❤❡ ♣r♦❜❧❡♠✿ ❣✐✈❡♥ (G, H) ∈ In t♦ ❝♦♥str✉❝t π ∈ Sn s✉❝❤

t❤❛t πG = H✳

◮ ●✐✈❡♥ ✐s♦♠♦r♣❤✐❝ ❣r❛♣❤s G, H ∈ Gn✱ t❤❡ ❛❧❣♦r✐t❤♠ ✜rst

❝♦♠♣✉t❡s t❤❡✐r t②♣❡✲✈❡❝t♦rs τG = (τ (g1) , . . . , τ (gn)) , τH = (τ (h1) , . . . , τ (hn)) . ■❢ t❤❡ ❝♦♦r❞✐♥❛t❡s ♦❢ ❛t ❧❡❛st ♦♥❡ ✈❡❝t♦r ❛r❡ ♥♦t ❞✐st✐♥❝t✱ t❤❡ ❛❧❣♦r✐t❤♠ ❛♥s✇❡rs ✧❄✧✭❞♦♥✬t ❦♥♦✇✮✳ ❖t❤❡r✇✐s❡ ❛r❡ ❜♦t❤ ♦❜❧✐q✉❡ ❛♥❞ ❝❤❡❝❦s ✇❤❡t❤❡r t❤❡ ♠❛♣ ✐s ❛♥ ✐s♦♠♦r♣❤✐s♠ ♦r ♥♦t✳ ■❢ t❤✐s ♠❛♣ t✉r♥s ♦✉t t♦ ❜❡ ❛♥ ✐s♦♠♦r♣❤✐s♠✱ t❤❡♥ t❤❡ ❛❧❣♦r✐t❤♠ ❣✐✈❡s ❛s t❤❡ ❛♥s✇❡r t♦ t❤❡ ♣r♦❜❧❡♠✳

❙❡❛r❝❤✐♥❣ ❢♦r ❛ ●r❛♣❤ ■s♦♠♦r♣❤✐s♠ ❆❧❣♦r✐t❤♠ ❢♦r ❙●■

❙●■✿ P❛rt✐❛❧ ♣♦❧②♥♦♠✐❛❧ ❛❧❣♦r✐t❤♠ A

◮ ■♥♣✉t s❡t✿ I =∪ In✱ ✇❤❡r❡

In = {(G, H) : G, H ∈ Gn, G ≃ H}✳

◮ ❚❤❡ ♣r♦❜❧❡♠✿ ❣✐✈❡♥ (G, H) ∈ In t♦ ❝♦♥str✉❝t π ∈ Sn s✉❝❤

t❤❛t πG = H✳

◮ ●✐✈❡♥ ✐s♦♠♦r♣❤✐❝ ❣r❛♣❤s G, H ∈ Gn✱ t❤❡ ❛❧❣♦r✐t❤♠ ✜rst

❝♦♠♣✉t❡s t❤❡✐r t②♣❡✲✈❡❝t♦rs τG = (τ (g1) , . . . , τ (gn)) , τH = (τ (h1) , . . . , τ (hn)) .

◮ ■❢ t❤❡ ❝♦♦r❞✐♥❛t❡s ♦❢ ❛t ❧❡❛st ♦♥❡ ✈❡❝t♦r ❛r❡ ♥♦t ❞✐st✐♥❝t✱ t❤❡

❛❧❣♦r✐t❤♠ ❛♥s✇❡rs ✧❄✧✭❞♦♥✬t ❦♥♦✇✮✳ ❖t❤❡r✇✐s❡ ❛r❡ ❜♦t❤ ♦❜❧✐q✉❡ ❛♥❞ ❝❤❡❝❦s ✇❤❡t❤❡r t❤❡ ♠❛♣ ✐s ❛♥ ✐s♦♠♦r♣❤✐s♠ ♦r ♥♦t✳ ■❢ t❤✐s ♠❛♣ t✉r♥s ♦✉t t♦ ❜❡ ❛♥ ✐s♦♠♦r♣❤✐s♠✱ t❤❡♥ t❤❡ ❛❧❣♦r✐t❤♠ ❣✐✈❡s ❛s t❤❡ ❛♥s✇❡r t♦ t❤❡ ♣r♦❜❧❡♠✳

❙❡❛r❝❤✐♥❣ ❢♦r ❛ ●r❛♣❤ ■s♦♠♦r♣❤✐s♠ ❆❧❣♦r✐t❤♠ ❢♦r ❙●■

❙●■✿ P❛rt✐❛❧ ♣♦❧②♥♦♠✐❛❧ ❛❧❣♦r✐t❤♠ A

◮ ■♥♣✉t s❡t✿ I =∪ In✱ ✇❤❡r❡

In = {(G, H) : G, H ∈ Gn, G ≃ H}✳

◮ ❚❤❡ ♣r♦❜❧❡♠✿ ❣✐✈❡♥ (G, H) ∈ In t♦ ❝♦♥str✉❝t π ∈ Sn s✉❝❤

t❤❛t πG = H✳

◮ ●✐✈❡♥ ✐s♦♠♦r♣❤✐❝ ❣r❛♣❤s G, H ∈ Gn✱ t❤❡ ❛❧❣♦r✐t❤♠ ✜rst

❝♦♠♣✉t❡s t❤❡✐r t②♣❡✲✈❡❝t♦rs τG = (τ (g1) , . . . , τ (gn)) , τH = (τ (h1) , . . . , τ (hn)) .

◮ ■❢ t❤❡ ❝♦♦r❞✐♥❛t❡s ♦❢ ❛t ❧❡❛st ♦♥❡ ✈❡❝t♦r ❛r❡ ♥♦t ❞✐st✐♥❝t✱ t❤❡

❛❧❣♦r✐t❤♠ ❛♥s✇❡rs ✧❄✧✭❞♦♥✬t ❦♥♦✇✮✳

◮ ❖t❤❡r✇✐s❡ G, H ❛r❡ ❜♦t❤ ♦❜❧✐q✉❡ ❛♥❞ A ❝❤❡❝❦s ✇❤❡t❤❡r t❤❡

♠❛♣ φ : gi → hi ✐s ❛♥ ✐s♦♠♦r♣❤✐s♠ ♦r ♥♦t✳ ■❢ t❤✐s ♠❛♣ t✉r♥s ♦✉t t♦ ❜❡ ❛♥ ✐s♦♠♦r♣❤✐s♠✱ t❤❡♥ t❤❡ ❛❧❣♦r✐t❤♠ ❣✐✈❡s ❛s t❤❡ ❛♥s✇❡r t♦ t❤❡ ♣r♦❜❧❡♠✳

❙❡❛r❝❤✐♥❣ ❢♦r ❛ ●r❛♣❤ ■s♦♠♦r♣❤✐s♠ ❆❧❣♦r✐t❤♠ ❢♦r ❙●■

❙●■✿ P❛rt✐❛❧ ♣♦❧②♥♦♠✐❛❧ ❛❧❣♦r✐t❤♠ A

◮ ■♥♣✉t s❡t✿ I =∪ In✱ ✇❤❡r❡

In = {(G, H) : G, H ∈ Gn, G ≃ H}✳

◮ ❚❤❡ ♣r♦❜❧❡♠✿ ❣✐✈❡♥ (G, H) ∈ In t♦ ❝♦♥str✉❝t π ∈ Sn s✉❝❤

t❤❛t πG = H✳

◮ ●✐✈❡♥ ✐s♦♠♦r♣❤✐❝ ❣r❛♣❤s G, H ∈ Gn✱ t❤❡ ❛❧❣♦r✐t❤♠ ✜rst

❝♦♠♣✉t❡s t❤❡✐r t②♣❡✲✈❡❝t♦rs τG = (τ (g1) , . . . , τ (gn)) , τH = (τ (h1) , . . . , τ (hn)) .

◮ ■❢ t❤❡ ❝♦♦r❞✐♥❛t❡s ♦❢ ❛t ❧❡❛st ♦♥❡ ✈❡❝t♦r ❛r❡ ♥♦t ❞✐st✐♥❝t✱ t❤❡

❛❧❣♦r✐t❤♠ ❛♥s✇❡rs ✧❄✧✭❞♦♥✬t ❦♥♦✇✮✳

◮ ❖t❤❡r✇✐s❡ G, H ❛r❡ ❜♦t❤ ♦❜❧✐q✉❡ ❛♥❞ A ❝❤❡❝❦s ✇❤❡t❤❡r t❤❡

♠❛♣ φ : gi → hi ✐s ❛♥ ✐s♦♠♦r♣❤✐s♠ ♦r ♥♦t✳

◮ ■❢ t❤✐s ♠❛♣ t✉r♥s ♦✉t t♦ ❜❡ ❛♥ ✐s♦♠♦r♣❤✐s♠✱ t❤❡♥ t❤❡ ❛❧❣♦r✐t❤♠

❣✐✈❡s φ ❛s t❤❡ ❛♥s✇❡r t♦ t❤❡ ♣r♦❜❧❡♠✳

❙❡❛r❝❤✐♥❣ ❢♦r ❛ ●r❛♣❤ ■s♦♠♦r♣❤✐s♠ ❆❧❣♦r✐t❤♠ ❢♦r ❙●■

❙●■✿ P❛rt✐❛❧ ♣♦❧②♥♦♠✐❛❧ ❛❧❣♦r✐t❤♠ A

◮ ■♥♣✉t s❡t✿ I =∪ In✱ ✇❤❡r❡

In = {(G, H) : G, H ∈ Gn, G ≃ H}✳

◮ ❚❤❡ ♣r♦❜❧❡♠✿ ❣✐✈❡♥ (G, H) ∈ In t♦ ❝♦♥str✉❝t π ∈ Sn s✉❝❤

t❤❛t πG = H✳

◮ ●✐✈❡♥ ✐s♦♠♦r♣❤✐❝ ❣r❛♣❤s G, H ∈ Gn✱ t❤❡ ❛❧❣♦r✐t❤♠ ✜rst

❝♦♠♣✉t❡s t❤❡✐r t②♣❡✲✈❡❝t♦rs τG = (τ (g1) , . . . , τ (gn)) , τH = (τ (h1) , . . . , τ (hn)) .

◮ ■❢ t❤❡ ❝♦♦r❞✐♥❛t❡s ♦❢ ❛t ❧❡❛st ♦♥❡ ✈❡❝t♦r ❛r❡ ♥♦t ❞✐st✐♥❝t✱ t❤❡

❛❧❣♦r✐t❤♠ ❛♥s✇❡rs ✧❄✧✭❞♦♥✬t ❦♥♦✇✮✳

◮ ❖t❤❡r✇✐s❡ G, H ❛r❡ ❜♦t❤ ♦❜❧✐q✉❡ ❛♥❞ A ❝❤❡❝❦s ✇❤❡t❤❡r t❤❡

♠❛♣ φ : gi → hi ✐s ❛♥ ✐s♦♠♦r♣❤✐s♠ ♦r ♥♦t✳

◮ ■❢ t❤✐s ♠❛♣ t✉r♥s ♦✉t t♦ ❜❡ ❛♥ ✐s♦♠♦r♣❤✐s♠✱ t❤❡♥ t❤❡ ❛❧❣♦r✐t❤♠

❣✐✈❡s φ ❛s t❤❡ ❛♥s✇❡r t♦ t❤❡ ♣r♦❜❧❡♠✳

❙❡❛r❝❤✐♥❣ ❢♦r ❛ ●r❛♣❤ ■s♦♠♦r♣❤✐s♠ ❆❧❣♦r✐t❤♠ ❢♦r ❙●■

❙●■✿ ❋❛✐❧✉r❡ ♣r♦❜❛❜✐❧✐t②

◮ A ❣✐✈❡s t❤❡ r✐❣❤t ❛♥s✇❡r ❢♦r t❤❡ s❡t ♦❢ ✐♥♣✉ts In ∩ (Go n × Go n)

❛♥❞ ❢❛✐❧s ♦♥ t❤❡ ❝♦♠♣❧❡♠❡♥t s❡t In ∩ (Sn × Sn) , ✇❤❡r❡ Sn = Gn − Go

n ✳

❚❤❡♦r❡♠

❚❤❡ ❢❛✐❧✉r❡ ♣r♦❜❛❜✐❧✐t② ♦❢ t❤❡ ❛❧❣♦r✐t❤♠ t❡♥❞s t♦ ③❡r♦ ❛s t❡♥❞s t♦ ✐♥✜♥✐t②✳ ■♥ ♦t❤❡r ✇♦r❞s✱ t❤❡ ❛❧❣♦r✐t❤♠ s♦❧✈❡s ❙●■ ✇✐t❤ ❛ ❤✐❣❤ ♣r♦❜❛❜✐❧✐t②✳ ❏✳❙❝❤r❡②❡r ✭✷✵✵✼✮✿ ❚❤❡ ♣r♦♣❡rt② ♦❢ ❛ ❣r❛♣❤ t♦ ❜❡ ♦❜❧✐q✉❡ ❤♦❧❞s ❛s②♠♣t♦t✐❝❛❧❧② ❛❧♠♦st s✉r❡❧②✱ ✐✳❡✳ ♦r✱ ❡q✉✐✈❛❧❡♥t❧②✱ ❲❡ ❤❛✈❡ t♦ ♣r♦✈❡ t❤❛t ✳

❙❡❛r❝❤✐♥❣ ❢♦r ❛ ●r❛♣❤ ■s♦♠♦r♣❤✐s♠ ❆❧❣♦r✐t❤♠ ❢♦r ❙●■

❙●■✿ ❋❛✐❧✉r❡ ♣r♦❜❛❜✐❧✐t②

◮ A ❣✐✈❡s t❤❡ r✐❣❤t ❛♥s✇❡r ❢♦r t❤❡ s❡t ♦❢ ✐♥♣✉ts In ∩ (Go n × Go n)

❛♥❞ ❢❛✐❧s ♦♥ t❤❡ ❝♦♠♣❧❡♠❡♥t s❡t In ∩ (Sn × Sn) , ✇❤❡r❡ Sn = Gn − Go

n ✳ ◮ ❚❤❡♦r❡♠

❚❤❡ ❢❛✐❧✉r❡ ♣r♦❜❛❜✐❧✐t② ♦❢ t❤❡ ❛❧❣♦r✐t❤♠ A t❡♥❞s t♦ ③❡r♦ ❛s n t❡♥❞s t♦ ✐♥✜♥✐t②✳ ■♥ ♦t❤❡r ✇♦r❞s✱ t❤❡ ❛❧❣♦r✐t❤♠ s♦❧✈❡s ❙●■ ✇✐t❤ ❛ ❤✐❣❤ ♣r♦❜❛❜✐❧✐t②✳ ❏✳❙❝❤r❡②❡r ✭✷✵✵✼✮✿ ❚❤❡ ♣r♦♣❡rt② ♦❢ ❛ ❣r❛♣❤ t♦ ❜❡ ♦❜❧✐q✉❡ ❤♦❧❞s ❛s②♠♣t♦t✐❝❛❧❧② ❛❧♠♦st s✉r❡❧②✱ ✐✳❡✳ ♦r✱ ❡q✉✐✈❛❧❡♥t❧②✱ ❲❡ ❤❛✈❡ t♦ ♣r♦✈❡ t❤❛t ✳

❙❡❛r❝❤✐♥❣ ❢♦r ❛ ●r❛♣❤ ■s♦♠♦r♣❤✐s♠ ❆❧❣♦r✐t❤♠ ❢♦r ❙●■

❙●■✿ ❋❛✐❧✉r❡ ♣r♦❜❛❜✐❧✐t②

◮ A ❣✐✈❡s t❤❡ r✐❣❤t ❛♥s✇❡r ❢♦r t❤❡ s❡t ♦❢ ✐♥♣✉ts In ∩ (Go n × Go n)

❛♥❞ ❢❛✐❧s ♦♥ t❤❡ ❝♦♠♣❧❡♠❡♥t s❡t In ∩ (Sn × Sn) , ✇❤❡r❡ Sn = Gn − Go

n ✳ ◮ ❚❤❡♦r❡♠

❚❤❡ ❢❛✐❧✉r❡ ♣r♦❜❛❜✐❧✐t② ♦❢ t❤❡ ❛❧❣♦r✐t❤♠ A t❡♥❞s t♦ ③❡r♦ ❛s n t❡♥❞s t♦ ✐♥✜♥✐t②✳ ■♥ ♦t❤❡r ✇♦r❞s✱ t❤❡ ❛❧❣♦r✐t❤♠ s♦❧✈❡s ❙●■ ✇✐t❤ ❛ ❤✐❣❤ ♣r♦❜❛❜✐❧✐t②✳

◮ ❏✳❙❝❤r❡②❡r ✭✷✵✵✼✮✿ ❚❤❡ ♣r♦♣❡rt② ♦❢ ❛ ❣r❛♣❤ G ∈ Gn t♦ ❜❡

♦❜❧✐q✉❡ ❤♦❧❞s ❛s②♠♣t♦t✐❝❛❧❧② ❛❧♠♦st s✉r❡❧②✱ ✐✳❡✳ lim

n→∞

|Go

n|

|Gn| = 1, ♦r✱ ❡q✉✐✈❛❧❡♥t❧②✱ lim

n→∞

|Sn| |Gn| = 1. ❲❡ ❤❛✈❡ t♦ ♣r♦✈❡ t❤❛t ✳

❙❡❛r❝❤✐♥❣ ❢♦r ❛ ●r❛♣❤ ■s♦♠♦r♣❤✐s♠ ❆❧❣♦r✐t❤♠ ❢♦r ❙●■

❙●■✿ ❋❛✐❧✉r❡ ♣r♦❜❛❜✐❧✐t②

◮ A ❣✐✈❡s t❤❡ r✐❣❤t ❛♥s✇❡r ❢♦r t❤❡ s❡t ♦❢ ✐♥♣✉ts In ∩ (Go n × Go n)

❛♥❞ ❢❛✐❧s ♦♥ t❤❡ ❝♦♠♣❧❡♠❡♥t s❡t In ∩ (Sn × Sn) , ✇❤❡r❡ Sn = Gn − Go

n ✳ ◮ ❚❤❡♦r❡♠

❚❤❡ ❢❛✐❧✉r❡ ♣r♦❜❛❜✐❧✐t② ♦❢ t❤❡ ❛❧❣♦r✐t❤♠ A t❡♥❞s t♦ ③❡r♦ ❛s n t❡♥❞s t♦ ✐♥✜♥✐t②✳ ■♥ ♦t❤❡r ✇♦r❞s✱ t❤❡ ❛❧❣♦r✐t❤♠ s♦❧✈❡s ❙●■ ✇✐t❤ ❛ ❤✐❣❤ ♣r♦❜❛❜✐❧✐t②✳

◮ ❏✳❙❝❤r❡②❡r ✭✷✵✵✼✮✿ ❚❤❡ ♣r♦♣❡rt② ♦❢ ❛ ❣r❛♣❤ G ∈ Gn t♦ ❜❡

♦❜❧✐q✉❡ ❤♦❧❞s ❛s②♠♣t♦t✐❝❛❧❧② ❛❧♠♦st s✉r❡❧②✱ ✐✳❡✳ lim

n→∞

|Go

n|

|Gn| = 1, ♦r✱ ❡q✉✐✈❛❧❡♥t❧②✱ lim

n→∞

|Sn| |Gn| = 1. ❲❡ ❤❛✈❡ t♦ ♣r♦✈❡ t❤❛t ✳

❙❡❛r❝❤✐♥❣ ❢♦r ❛ ●r❛♣❤ ■s♦♠♦r♣❤✐s♠ ❆❧❣♦r✐t❤♠ ❢♦r ❙●■

❙●■✿ ❋❛✐❧✉r❡ ♣r♦❜❛❜✐❧✐t②

◮ A ❣✐✈❡s t❤❡ r✐❣❤t ❛♥s✇❡r ❢♦r t❤❡ s❡t ♦❢ ✐♥♣✉ts In ∩ (Go n × Go n)

❛♥❞ ❢❛✐❧s ♦♥ t❤❡ ❝♦♠♣❧❡♠❡♥t s❡t In ∩ (Sn × Sn) , ✇❤❡r❡ Sn = Gn − Go

n ✳ ◮ ❚❤❡♦r❡♠

❚❤❡ ❢❛✐❧✉r❡ ♣r♦❜❛❜✐❧✐t② ♦❢ t❤❡ ❛❧❣♦r✐t❤♠ A t❡♥❞s t♦ ③❡r♦ ❛s n t❡♥❞s t♦ ✐♥✜♥✐t②✳ ■♥ ♦t❤❡r ✇♦r❞s✱ t❤❡ ❛❧❣♦r✐t❤♠ s♦❧✈❡s ❙●■ ✇✐t❤ ❛ ❤✐❣❤ ♣r♦❜❛❜✐❧✐t②✳

◮ ❏✳❙❝❤r❡②❡r ✭✷✵✵✼✮✿ ❚❤❡ ♣r♦♣❡rt② ♦❢ ❛ ❣r❛♣❤ G ∈ Gn t♦ ❜❡

♦❜❧✐q✉❡ ❤♦❧❞s ❛s②♠♣t♦t✐❝❛❧❧② ❛❧♠♦st s✉r❡❧②✱ ✐✳❡✳ lim

n→∞

|Go

n|

|Gn| = 1, ♦r✱ ❡q✉✐✈❛❧❡♥t❧②✱ lim

n→∞

|Sn| |Gn| = 1. ❲❡ ❤❛✈❡ t♦ ♣r♦✈❡ t❤❛t ✳

❙❡❛r❝❤✐♥❣ ❢♦r ❛ ●r❛♣❤ ■s♦♠♦r♣❤✐s♠ ❆❧❣♦r✐t❤♠ ❢♦r ❙●■

❙●■✿ ❋❛✐❧✉r❡ ♣r♦❜❛❜✐❧✐t②

◮ ❈♦♥s✐❞❡r pr1 : In → Gn✳

❲❤❛t ❛r❡ t❤❡ ♣r❡✐♠❛❣❡s ♦❢ ❄ ❢♦r ❡✈❡r② ✳ ■♥ ❝❛s❡ ✐s ♦❜❧✐q✉❡ ✇❡ ❝❛♥ s❛② ♠♦r❡✿ ■♥❞❡❡❞✱ ❛♥❞ t❤✉s t❤❡ ♦r❜✐t ❝♦♥s✐sts ♦❢ ❣r❛♣❤s ❛♥❞ t❤❡ s❡t ✐s t❤❡ ♣r❡✐♠❛❣❡ ♦❢ ✉♥❞❡r ✳ ❈♦♥❝❧✉❞❡✿ ❤❛s t❤❡ ✜❜❡rs ♦❢ ❝❛r❞✐♥❛❧✐t② ✱ ❤❡♥❝❡ ❛♥❞ s♦

❙❡❛r❝❤✐♥❣ ❢♦r ❛ ●r❛♣❤ ■s♦♠♦r♣❤✐s♠ ❆❧❣♦r✐t❤♠ ❢♦r ❙●■

❙●■✿ ❋❛✐❧✉r❡ ♣r♦❜❛❜✐❧✐t②

◮ ❈♦♥s✐❞❡r pr1 : In → Gn✳ ◮ ❲❤❛t ❛r❡ t❤❡ ♣r❡✐♠❛❣❡s ♦❢ pr1❄

❢♦r ❡✈❡r② ✳ ■♥ ❝❛s❡ ✐s ♦❜❧✐q✉❡ ✇❡ ❝❛♥ s❛② ♠♦r❡✿ ■♥❞❡❡❞✱ ❛♥❞ t❤✉s t❤❡ ♦r❜✐t ❝♦♥s✐sts ♦❢ ❣r❛♣❤s ❛♥❞ t❤❡ s❡t ✐s t❤❡ ♣r❡✐♠❛❣❡ ♦❢ ✉♥❞❡r ✳ ❈♦♥❝❧✉❞❡✿ ❤❛s t❤❡ ✜❜❡rs ♦❢ ❝❛r❞✐♥❛❧✐t② ✱ ❤❡♥❝❡ ❛♥❞ s♦

❙❡❛r❝❤✐♥❣ ❢♦r ❛ ●r❛♣❤ ■s♦♠♦r♣❤✐s♠ ❆❧❣♦r✐t❤♠ ❢♦r ❙●■

❙●■✿ ❋❛✐❧✉r❡ ♣r♦❜❛❜✐❧✐t②

◮ ❈♦♥s✐❞❡r pr1 : In → Gn✳ ◮ ❲❤❛t ❛r❡ t❤❡ ♣r❡✐♠❛❣❡s ♦❢ pr1❄ ◮

pr−1

1 G

• ≤ n! ❢♦r ❡✈❡r② G ∈ Gn✳

■♥ ❝❛s❡ ✐s ♦❜❧✐q✉❡ ✇❡ ❝❛♥ s❛② ♠♦r❡✿ ■♥❞❡❡❞✱ ❛♥❞ t❤✉s t❤❡ ♦r❜✐t ❝♦♥s✐sts ♦❢ ❣r❛♣❤s ❛♥❞ t❤❡ s❡t ✐s t❤❡ ♣r❡✐♠❛❣❡ ♦❢ ✉♥❞❡r ✳ ❈♦♥❝❧✉❞❡✿ ❤❛s t❤❡ ✜❜❡rs ♦❢ ❝❛r❞✐♥❛❧✐t② ✱ ❤❡♥❝❡ ❛♥❞ s♦

❙❡❛r❝❤✐♥❣ ❢♦r ❛ ●r❛♣❤ ■s♦♠♦r♣❤✐s♠ ❆❧❣♦r✐t❤♠ ❢♦r ❙●■

❙●■✿ ❋❛✐❧✉r❡ ♣r♦❜❛❜✐❧✐t②

◮ ❈♦♥s✐❞❡r pr1 : In → Gn✳ ◮ ❲❤❛t ❛r❡ t❤❡ ♣r❡✐♠❛❣❡s ♦❢ pr1❄ ◮

pr−1

1 G

• ≤ n! ❢♦r ❡✈❡r② G ∈ Gn✳

◮ ■♥ ❝❛s❡ G ✐s ♦❜❧✐q✉❡ ✇❡ ❝❛♥ s❛② ♠♦r❡✿

• pr−1

1 G

• = n!

■♥❞❡❡❞✱ ❛♥❞ t❤✉s t❤❡ ♦r❜✐t ❝♦♥s✐sts ♦❢ ❣r❛♣❤s ❛♥❞ t❤❡ s❡t ✐s t❤❡ ♣r❡✐♠❛❣❡ ♦❢ ✉♥❞❡r ✳ ❈♦♥❝❧✉❞❡✿ ❤❛s t❤❡ ✜❜❡rs ♦❢ ❝❛r❞✐♥❛❧✐t② ✱ ❤❡♥❝❡ ❛♥❞ s♦

❙❡❛r❝❤✐♥❣ ❢♦r ❛ ●r❛♣❤ ■s♦♠♦r♣❤✐s♠ ❆❧❣♦r✐t❤♠ ❢♦r ❙●■

❙●■✿ ❋❛✐❧✉r❡ ♣r♦❜❛❜✐❧✐t②

◮ ❈♦♥s✐❞❡r pr1 : In → Gn✳ ◮ ❲❤❛t ❛r❡ t❤❡ ♣r❡✐♠❛❣❡s ♦❢ pr1❄ ◮

pr−1

1 G

• ≤ n! ❢♦r ❡✈❡r② G ∈ Gn✳

◮ ■♥ ❝❛s❡ G ✐s ♦❜❧✐q✉❡ ✇❡ ❝❛♥ s❛② ♠♦r❡✿

• pr−1

1 G

• = n!

◮ ■♥❞❡❡❞✱ Aut (G) = 1 ❛♥❞ t❤✉s t❤❡ ♦r❜✐t SnG ❝♦♥s✐sts ♦❢ n!

❣r❛♣❤s ❛♥❞ t❤❡ s❡t (G, SnG) ✐s t❤❡ ♣r❡✐♠❛❣❡ ♦❢ G ✉♥❞❡r pr1✳ ❈♦♥❝❧✉❞❡✿ ❤❛s t❤❡ ✜❜❡rs ♦❢ ❝❛r❞✐♥❛❧✐t② ✱ ❤❡♥❝❡ ❛♥❞ s♦

❙❡❛r❝❤✐♥❣ ❢♦r ❛ ●r❛♣❤ ■s♦♠♦r♣❤✐s♠ ❆❧❣♦r✐t❤♠ ❢♦r ❙●■

❙●■✿ ❋❛✐❧✉r❡ ♣r♦❜❛❜✐❧✐t②

◮ ❈♦♥s✐❞❡r pr1 : In → Gn✳ ◮ ❲❤❛t ❛r❡ t❤❡ ♣r❡✐♠❛❣❡s ♦❢ pr1❄ ◮

pr−1

1 G

• ≤ n! ❢♦r ❡✈❡r② G ∈ Gn✳

◮ ■♥ ❝❛s❡ G ✐s ♦❜❧✐q✉❡ ✇❡ ❝❛♥ s❛② ♠♦r❡✿

• pr−1

1 G

• = n!

◮ ■♥❞❡❡❞✱ Aut (G) = 1 ❛♥❞ t❤✉s t❤❡ ♦r❜✐t SnG ❝♦♥s✐sts ♦❢ n!

❣r❛♣❤s ❛♥❞ t❤❡ s❡t (G, SnG) ✐s t❤❡ ♣r❡✐♠❛❣❡ ♦❢ G ✉♥❞❡r pr1✳

◮ ❈♦♥❝❧✉❞❡✿ pr1 : In ∩ (Go n × Go n) → Go n ❤❛s t❤❡ ✜❜❡rs ♦❢

❝❛r❞✐♥❛❧✐t② n!✱ ❤❡♥❝❡ |In ∩ (Go

n × Go n)| = n! |Go n| ❛♥❞ s♦

|In| ≥ n! |Go

n| .

❙❡❛r❝❤✐♥❣ ❢♦r ❛ ●r❛♣❤ ■s♦♠♦r♣❤✐s♠ ❆❧❣♦r✐t❤♠ ❢♦r ❙●■

❙●■✿ ❋❛✐❧✉r❡ ♣r♦❜❛❜✐❧✐t②

◮ ❈♦♥s✐❞❡r pr1 : In → Gn✳ ◮ ❲❤❛t ❛r❡ t❤❡ ♣r❡✐♠❛❣❡s ♦❢ pr1❄ ◮

pr−1

1 G

• ≤ n! ❢♦r ❡✈❡r② G ∈ Gn✳

◮ ■♥ ❝❛s❡ G ✐s ♦❜❧✐q✉❡ ✇❡ ❝❛♥ s❛② ♠♦r❡✿

• pr−1

1 G

• = n!

◮ ■♥❞❡❡❞✱ Aut (G) = 1 ❛♥❞ t❤✉s t❤❡ ♦r❜✐t SnG ❝♦♥s✐sts ♦❢ n!

❣r❛♣❤s ❛♥❞ t❤❡ s❡t (G, SnG) ✐s t❤❡ ♣r❡✐♠❛❣❡ ♦❢ G ✉♥❞❡r pr1✳

◮ ❈♦♥❝❧✉❞❡✿ pr1 : In ∩ (Go n × Go n) → Go n ❤❛s t❤❡ ✜❜❡rs ♦❢

❝❛r❞✐♥❛❧✐t② n!✱ ❤❡♥❝❡ |In ∩ (Go

n × Go n)| = n! |Go n| ❛♥❞ s♦

|In| ≥ n! |Go

n| .

❙❡❛r❝❤✐♥❣ ❢♦r ❛ ●r❛♣❤ ■s♦♠♦r♣❤✐s♠ ❆❧❣♦r✐t❤♠ ❢♦r ❙●■

❙●■✿ ❋❛✐❧✉r❡ ♣r♦❜❛❜✐❧✐t②

◮ ❈♦♥s✐❞❡r✐♥❣ t❤❡ ✜❜❡rs ♦❢ pr1 ♦✈❡r t❤❡ s❡t Sn ✇❡ ❝♦♥❝❧✉❞❡ t❤❛t

|In ∩ (Sn × Sn)| ≤ n! |Sn| .

❙❡❛r❝❤✐♥❣ ❢♦r ❛ ●r❛♣❤ ■s♦♠♦r♣❤✐s♠ ❆❧❣♦r✐t❤♠ ❢♦r ❙●■

❙●■✿ ❋❛✐❧✉r❡ ♣r♦❜❛❜✐❧✐t②

◮ ❈♦♥s✐❞❡r✐♥❣ t❤❡ ✜❜❡rs ♦❢ pr1 ♦✈❡r t❤❡ s❡t Sn ✇❡ ❝♦♥❝❧✉❞❡ t❤❛t

|In ∩ (Sn × Sn)| ≤ n! |Sn| .

◮

|In ∩ (Sn × Sn)| |In| ≤ n! |Sn| n! |Go

n| = |Sn|

|Go

n| =

|Sn| |Gn| − |Sn|

❙❡❛r❝❤✐♥❣ ❢♦r ❛ ●r❛♣❤ ■s♦♠♦r♣❤✐s♠ ❆❧❣♦r✐t❤♠ ❢♦r ❙●■

❙●■✿ ❋❛✐❧✉r❡ ♣r♦❜❛❜✐❧✐t②

◮ ❈♦♥s✐❞❡r✐♥❣ t❤❡ ✜❜❡rs ♦❢ pr1 ♦✈❡r t❤❡ s❡t Sn ✇❡ ❝♦♥❝❧✉❞❡ t❤❛t

|In ∩ (Sn × Sn)| ≤ n! |Sn| .

◮

|In ∩ (Sn × Sn)| |In| ≤ n! |Sn| n! |Go

n| = |Sn|

|Go

n| =

|Sn| |Gn| − |Sn|

◮

|Sn| |Gn|   1 1 − |Sn|

|Gn|

  → 0, n → ∞.

❙❡❛r❝❤✐♥❣ ❢♦r ❛ ●r❛♣❤ ■s♦♠♦r♣❤✐s♠ ❆❧❣♦r✐t❤♠ ❢♦r ❙●■

❙●■✿ ❋❛✐❧✉r❡ ♣r♦❜❛❜✐❧✐t②

◮ ❈♦♥s✐❞❡r✐♥❣ t❤❡ ✜❜❡rs ♦❢ pr1 ♦✈❡r t❤❡ s❡t Sn ✇❡ ❝♦♥❝❧✉❞❡ t❤❛t

|In ∩ (Sn × Sn)| ≤ n! |Sn| .

◮

|In ∩ (Sn × Sn)| |In| ≤ n! |Sn| n! |Go

n| = |Sn|

|Go

n| =

|Sn| |Gn| − |Sn|

◮

|Sn| |Gn|   1 1 − |Sn|

|Gn|

  → 0, n → ∞.

❙❡❛r❝❤✐♥❣ ❢♦r ❛ ●r❛♣❤ ■s♦♠♦r♣❤✐s♠ ❉✐s❡❛s❡

❉✐s❡❛s❡

❘❡❛❞✱ ❘♦♥❛❧❞ ❈✳❀ ❈♦r♥❡✐❧✱ ❉❡r❡❦ ●✳ ✭✶✾✼✼✮✱ ❚❤❡ ❣r❛♣❤ ✐s♦♠♦r♣❤✐s♠ ❞✐s❡❛s❡✱ ❏♦✉r♥❛❧ ♦❢ ●r❛♣❤ ❚❤❡♦r② ✶ ✭✹✮✿ ✸✸✾✲✸✻✸✱

❙❡❛r❝❤✐♥❣ ❢♦r ❛ ●r❛♣❤ ■s♦♠♦r♣❤✐s♠ ❉✐s❡❛s❡

❉✐s❡❛s❡

❘❡❛❞✱ ❘♦♥❛❧❞ ❈✳❀ ❈♦r♥❡✐❧✱ ❉❡r❡❦ ●✳ ✭✶✾✼✼✮✱ ❚❤❡ ❣r❛♣❤ ✐s♦♠♦r♣❤✐s♠ ❞✐s❡❛s❡✱ ❏♦✉r♥❛❧ ♦❢ ●r❛♣❤ ❚❤❡♦r② ✶ ✭✹✮✿ ✸✸✾✲✸✻✸✱

❙❡❛r❝❤✐♥❣ ❢♦r ❛ ●r❛♣❤ ■s♦♠♦r♣❤✐s♠ ❉✐s❡❛s❡

❉✐s❡❛s❡

❘❡❛❞✱ ❘♦♥❛❧❞ ❈✳❀ ❈♦r♥❡✐❧✱ ❉❡r❡❦ ●✳ ✭✶✾✼✼✮✱ ❚❤❡ ❣r❛♣❤ ✐s♦♠♦r♣❤✐s♠ ❞✐s❡❛s❡✱ ❏♦✉r♥❛❧ ♦❢ ●r❛♣❤ ❚❤❡♦r② ✶ ✭✹✮✿ ✸✸✾✲✸✻✸✱

❙❡❛r❝❤✐♥❣ ❢♦r ❛ ●r❛♣❤ ■s♦♠♦r♣❤✐s♠ ❉✐s❡❛s❡

❉✐s❡❛s❡

❘❡❛❞✱ ❘♦♥❛❧❞ ❈✳❀ ❈♦r♥❡✐❧✱ ❉❡r❡❦ ●✳ ✭✶✾✼✼✮✱ ❚❤❡ ❣r❛♣❤ ✐s♦♠♦r♣❤✐s♠ ❞✐s❡❛s❡✱ ❏♦✉r♥❛❧ ♦❢ ●r❛♣❤ ❚❤❡♦r② ✶ ✭✹✮✿ ✸✸✾✲✸✻✸✱

❙❡❛r❝❤✐♥❣ ❢♦r ❛ ●r❛♣❤ ■s♦♠♦r♣❤✐s♠ ❉✐s❡❛s❡

❙✐❣♥s t❤❛t ❛ ♣r♦❜❧❡♠ ✐s ❛ ▼❛t❤❡♠❛t✐❝❛❧ ❉✐s❡❛s❡ ✭❘✳▲✐♣t♦♥✮

◮ ✶✳ ❆ ♣r♦❜❧❡♠ ♠✉st ❜❡ ❡❛s② t♦ st❛t❡ t♦ ❜❡ ❛ ▼❉✳

❍♦❞❣❡✲❈♦♥❥❡❝t✉r❡ ✭106 ❯❙❉✮ ✇✐❧❧ ♥❡✈❡r ❜❡ ❛ ❞✐s❡❛s❡✳ ✷✳ ❆ ♣r♦❜❧❡♠ ♠✉st s❡❡♠ t♦ ❜❡ ❛❝❝❡ss✐❜❧❡✱ ❡✈❡♥ t♦ ❛♥ ❛♠❛t❡✉r✳ ❨♦✉r r❡❛❝t✐♦♥ s❤♦✉❧❞ ❜❡✿ t❤❛t ✐s ♦♣❡♥❄ ✸✳ ❚❤❡ ♣r♦❜❧❡♠ ♠✉st s❡❡♠ t♦ ❜❡ ❡❛s②✳ ❆ ♣r♦❜❧❡♠ ♠✉st ❛❧s♦ ❤❛✈❡ ❜❡❡♥ r❡♣❡❛t❡❞❧② ✧s♦❧✈❡❞✧t♦ ❜❡ ❛ tr✉❡ ▼❉✳ ❆ ❣♦♦❞ ▼❉ ✉s✉❛❧❧② ❤❛s ❜❡❡♥ ✧♣r♦✈❡❞✧♠❛♥② t✐♠❡s ✕ ♦❢t❡♥ ❜② t❤❡ s❛♠❡ ♣❡rs♦♥✳ ■❢ ②♦✉ s❡❡ ❛ ♣❛♣❡r ✐♥ ❛r❳✐✈✳♦r❣ ✇✐t❤ ♠❛♥② ✧✉♣❞❛t❡s✧t❤❛t ✐s ❛ ❣♦♦❞ s✐❣♥ t❤❛t t❤❡ ♣r♦❜❧❡♠ ✐s ❛ ▼❉✳

❙❡❛r❝❤✐♥❣ ❢♦r ❛ ●r❛♣❤ ■s♦♠♦r♣❤✐s♠ ❉✐s❡❛s❡

❙✐❣♥s t❤❛t ❛ ♣r♦❜❧❡♠ ✐s ❛ ▼❛t❤❡♠❛t✐❝❛❧ ❉✐s❡❛s❡ ✭❘✳▲✐♣t♦♥✮

◮ ✶✳ ❆ ♣r♦❜❧❡♠ ♠✉st ❜❡ ❡❛s② t♦ st❛t❡ t♦ ❜❡ ❛ ▼❉✳

❍♦❞❣❡✲❈♦♥❥❡❝t✉r❡ ✭106 ❯❙❉✮ ✇✐❧❧ ♥❡✈❡r ❜❡ ❛ ❞✐s❡❛s❡✳

◮ ✷✳ ❆ ♣r♦❜❧❡♠ ♠✉st s❡❡♠ t♦ ❜❡ ❛❝❝❡ss✐❜❧❡✱ ❡✈❡♥ t♦ ❛♥ ❛♠❛t❡✉r✳

❨♦✉r r❡❛❝t✐♦♥ s❤♦✉❧❞ ❜❡✿ t❤❛t ✐s ♦♣❡♥❄ ✸✳ ❚❤❡ ♣r♦❜❧❡♠ ♠✉st s❡❡♠ t♦ ❜❡ ❡❛s②✳ ❆ ♣r♦❜❧❡♠ ♠✉st ❛❧s♦ ❤❛✈❡ ❜❡❡♥ r❡♣❡❛t❡❞❧② ✧s♦❧✈❡❞✧t♦ ❜❡ ❛ tr✉❡ ▼❉✳ ❆ ❣♦♦❞ ▼❉ ✉s✉❛❧❧② ❤❛s ❜❡❡♥ ✧♣r♦✈❡❞✧♠❛♥② t✐♠❡s ✕ ♦❢t❡♥ ❜② t❤❡ s❛♠❡ ♣❡rs♦♥✳ ■❢ ②♦✉ s❡❡ ❛ ♣❛♣❡r ✐♥ ❛r❳✐✈✳♦r❣ ✇✐t❤ ♠❛♥② ✧✉♣❞❛t❡s✧t❤❛t ✐s ❛ ❣♦♦❞ s✐❣♥ t❤❛t t❤❡ ♣r♦❜❧❡♠ ✐s ❛ ▼❉✳

❙❡❛r❝❤✐♥❣ ❢♦r ❛ ●r❛♣❤ ■s♦♠♦r♣❤✐s♠ ❉✐s❡❛s❡

❙✐❣♥s t❤❛t ❛ ♣r♦❜❧❡♠ ✐s ❛ ▼❛t❤❡♠❛t✐❝❛❧ ❉✐s❡❛s❡ ✭❘✳▲✐♣t♦♥✮

◮ ✶✳ ❆ ♣r♦❜❧❡♠ ♠✉st ❜❡ ❡❛s② t♦ st❛t❡ t♦ ❜❡ ❛ ▼❉✳

❍♦❞❣❡✲❈♦♥❥❡❝t✉r❡ ✭106 ❯❙❉✮ ✇✐❧❧ ♥❡✈❡r ❜❡ ❛ ❞✐s❡❛s❡✳

◮ ✷✳ ❆ ♣r♦❜❧❡♠ ♠✉st s❡❡♠ t♦ ❜❡ ❛❝❝❡ss✐❜❧❡✱ ❡✈❡♥ t♦ ❛♥ ❛♠❛t❡✉r✳

❨♦✉r r❡❛❝t✐♦♥ s❤♦✉❧❞ ❜❡✿ t❤❛t ✐s ♦♣❡♥❄

◮ ✸✳ ❚❤❡ ♣r♦❜❧❡♠ ♠✉st s❡❡♠ t♦ ❜❡ ❡❛s②✳ ❆ ♣r♦❜❧❡♠ ♠✉st ❛❧s♦

❤❛✈❡ ❜❡❡♥ r❡♣❡❛t❡❞❧② ✧s♦❧✈❡❞✧t♦ ❜❡ ❛ tr✉❡ ▼❉✳ ❆ ❣♦♦❞ ▼❉ ✉s✉❛❧❧② ❤❛s ❜❡❡♥ ✧♣r♦✈❡❞✧♠❛♥② t✐♠❡s ✕ ♦❢t❡♥ ❜② t❤❡ s❛♠❡ ♣❡rs♦♥✳ ■❢ ②♦✉ s❡❡ ❛ ♣❛♣❡r ✐♥ ❛r❳✐✈✳♦r❣ ✇✐t❤ ♠❛♥② ✧✉♣❞❛t❡s✧t❤❛t ✐s ❛ ❣♦♦❞ s✐❣♥ t❤❛t t❤❡ ♣r♦❜❧❡♠ ✐s ❛ ▼❉✳

❙❡❛r❝❤✐♥❣ ❢♦r ❛ ●r❛♣❤ ■s♦♠♦r♣❤✐s♠ ❉✐s❡❛s❡

❙✐❣♥s t❤❛t ❛ ♣r♦❜❧❡♠ ✐s ❛ ▼❛t❤❡♠❛t✐❝❛❧ ❉✐s❡❛s❡ ✭❘✳▲✐♣t♦♥✮

◮ ✶✳ ❆ ♣r♦❜❧❡♠ ♠✉st ❜❡ ❡❛s② t♦ st❛t❡ t♦ ❜❡ ❛ ▼❉✳

❍♦❞❣❡✲❈♦♥❥❡❝t✉r❡ ✭106 ❯❙❉✮ ✇✐❧❧ ♥❡✈❡r ❜❡ ❛ ❞✐s❡❛s❡✳

◮ ✷✳ ❆ ♣r♦❜❧❡♠ ♠✉st s❡❡♠ t♦ ❜❡ ❛❝❝❡ss✐❜❧❡✱ ❡✈❡♥ t♦ ❛♥ ❛♠❛t❡✉r✳

❨♦✉r r❡❛❝t✐♦♥ s❤♦✉❧❞ ❜❡✿ t❤❛t ✐s ♦♣❡♥❄

◮ ✸✳ ❚❤❡ ♣r♦❜❧❡♠ ♠✉st s❡❡♠ t♦ ❜❡ ❡❛s②✳ ❆ ♣r♦❜❧❡♠ ♠✉st ❛❧s♦

❤❛✈❡ ❜❡❡♥ r❡♣❡❛t❡❞❧② ✧s♦❧✈❡❞✧t♦ ❜❡ ❛ tr✉❡ ▼❉✳ ❆ ❣♦♦❞ ▼❉ ✉s✉❛❧❧② ❤❛s ❜❡❡♥ ✧♣r♦✈❡❞✧♠❛♥② t✐♠❡s ✕ ♦❢t❡♥ ❜② t❤❡ s❛♠❡ ♣❡rs♦♥✳

◮ ■❢ ②♦✉ s❡❡ ❛ ♣❛♣❡r ✐♥ ❛r❳✐✈✳♦r❣ ✇✐t❤ ♠❛♥② ✧✉♣❞❛t❡s✧t❤❛t ✐s ❛

❣♦♦❞ s✐❣♥ t❤❛t t❤❡ ♣r♦❜❧❡♠ ✐s ❛ ▼❉✳

❙❡❛r❝❤✐♥❣ ❢♦r ❛ ●r❛♣❤ ■s♦♠♦r♣❤✐s♠ ❉✐s❡❛s❡

❙✐❣♥s t❤❛t ❛ ♣r♦❜❧❡♠ ✐s ❛ ▼❛t❤❡♠❛t✐❝❛❧ ❉✐s❡❛s❡ ✭❘✳▲✐♣t♦♥✮

◮ ✶✳ ❆ ♣r♦❜❧❡♠ ♠✉st ❜❡ ❡❛s② t♦ st❛t❡ t♦ ❜❡ ❛ ▼❉✳

❍♦❞❣❡✲❈♦♥❥❡❝t✉r❡ ✭106 ❯❙❉✮ ✇✐❧❧ ♥❡✈❡r ❜❡ ❛ ❞✐s❡❛s❡✳

◮ ✷✳ ❆ ♣r♦❜❧❡♠ ♠✉st s❡❡♠ t♦ ❜❡ ❛❝❝❡ss✐❜❧❡✱ ❡✈❡♥ t♦ ❛♥ ❛♠❛t❡✉r✳

❨♦✉r r❡❛❝t✐♦♥ s❤♦✉❧❞ ❜❡✿ t❤❛t ✐s ♦♣❡♥❄

◮ ✸✳ ❚❤❡ ♣r♦❜❧❡♠ ♠✉st s❡❡♠ t♦ ❜❡ ❡❛s②✳ ❆ ♣r♦❜❧❡♠ ♠✉st ❛❧s♦

❤❛✈❡ ❜❡❡♥ r❡♣❡❛t❡❞❧② ✧s♦❧✈❡❞✧t♦ ❜❡ ❛ tr✉❡ ▼❉✳ ❆ ❣♦♦❞ ▼❉ ✉s✉❛❧❧② ❤❛s ❜❡❡♥ ✧♣r♦✈❡❞✧♠❛♥② t✐♠❡s ✕ ♦❢t❡♥ ❜② t❤❡ s❛♠❡ ♣❡rs♦♥✳

◮ ■❢ ②♦✉ s❡❡ ❛ ♣❛♣❡r ✐♥ ❛r❳✐✈✳♦r❣ ✇✐t❤ ♠❛♥② ✧✉♣❞❛t❡s✧t❤❛t ✐s ❛

❣♦♦❞ s✐❣♥ t❤❛t t❤❡ ♣r♦❜❧❡♠ ✐s ❛ ▼❉✳

❙❡❛r❝❤✐♥❣ ❢♦r ❛ ●r❛♣❤ ■s♦♠♦r♣❤✐s♠ ❆ ❣✐❢t

• ■ ❛s ❛ ❣✐❢t

◮ ❉♦❡s P❂◆P❄ ❙✳❙♠❛❧❡✿ ■ s♦♠❡t✐♠❡s ❝♦♥s✐❞❡r t❤✐s ♣r♦❜❧❡♠ ❛s ❛

❣✐❢t t♦ ♠❛t❤❡♠❛t✐❝s ❢r♦♠ ❝♦♠♣✉t❡r s❝✐❡♥❝❡✳ ❉✐♦♣❤❛♥t ✿ ●✐✈❡♥ ✭❛s ✐♥♣✉t✮ ♣♦❧②♥♦♠✐❛❧s ✐♥ ✈❛r✐❛❜❧❡s ✇✐t❤ ❝♦❡✣❝✐❡♥ts ✐♥ ✳ ❉❡❝✐❞❡ ✐❢ t❤❡r❡ ✐s ❛ ❝♦♠♠♦♥ ③❡r♦ ❄ ❈♦♥❥❡❝t✉r❡✳ ❚❤❡r❡ ✐s ♥♦ ♣♦❧②♥♦♠✐❛❧ t✐♠❡ ❛❧❣♦r✐t❤♠ ♦✈❡r ❞❡❝✐❞✐♥❣ t❤✐s ♣r♦❜❧❡♠✳ ❚❤✐s ✐s ❛ r❡❢♦r♠✉❧❛t✐♦♥ ♦❢ t❤❡ ❝❧❛ss✐❝ ❝♦♥❥❡❝t✉r❡ ✳ ◗✉❡st✐♦♥s✳ ❲❤❛t ✐s t❤❡ r❡❞✉❝t✐♦♥ r❡❧❛t✐♦♥ ❜❡t✇❡❡♥ ●■ ❛♥❞ ❄ ❙♣❡❝✐✜❝❛❧❧②✱ ❝❛♥ ❜❡ ♣♦❧②♥♦♠✐❛❧❧② r❡❞✉❝❡❞ t♦ ❄ ■s ♣♦❧②♥♦♠✐❛❧❧② r❡❞✉❝✐❜❧❡ t♦ ❄

❙❡❛r❝❤✐♥❣ ❢♦r ❛ ●r❛♣❤ ■s♦♠♦r♣❤✐s♠ ❆ ❣✐❢t

• ■ ❛s ❛ ❣✐❢t

◮ ❉♦❡s P❂◆P❄ ❙✳❙♠❛❧❡✿ ■ s♦♠❡t✐♠❡s ❝♦♥s✐❞❡r t❤✐s ♣r♦❜❧❡♠ ❛s ❛

❣✐❢t t♦ ♠❛t❤❡♠❛t✐❝s ❢r♦♠ ❝♦♠♣✉t❡r s❝✐❡♥❝❡✳

◮ ❉✐♦♣❤❛♥t(Z/2)✿ ●✐✈❡♥ ✭❛s ✐♥♣✉t✮ k ♣♦❧②♥♦♠✐❛❧s ✐♥ n ✈❛r✐❛❜❧❡s

✇✐t❤ ❝♦❡✣❝✐❡♥ts ✐♥ Z/2✳ ❉❡❝✐❞❡ ✐❢ t❤❡r❡ ✐s ❛ ❝♦♠♠♦♥ ③❡r♦ x ∈ (Z/2)n❄ ❈♦♥❥❡❝t✉r❡✳ ❚❤❡r❡ ✐s ♥♦ ♣♦❧②♥♦♠✐❛❧ t✐♠❡ ❛❧❣♦r✐t❤♠ ♦✈❡r ❞❡❝✐❞✐♥❣ t❤✐s ♣r♦❜❧❡♠✳ ❚❤✐s ✐s ❛ r❡❢♦r♠✉❧❛t✐♦♥ ♦❢ t❤❡ ❝❧❛ss✐❝ ❝♦♥❥❡❝t✉r❡ ✳ ◗✉❡st✐♦♥s✳ ❲❤❛t ✐s t❤❡ r❡❞✉❝t✐♦♥ r❡❧❛t✐♦♥ ❜❡t✇❡❡♥ ●■ ❛♥❞ ❄ ❙♣❡❝✐✜❝❛❧❧②✱ ❝❛♥ ❜❡ ♣♦❧②♥♦♠✐❛❧❧② r❡❞✉❝❡❞ t♦ ❄ ■s ♣♦❧②♥♦♠✐❛❧❧② r❡❞✉❝✐❜❧❡ t♦ ❄

❙❡❛r❝❤✐♥❣ ❢♦r ❛ ●r❛♣❤ ■s♦♠♦r♣❤✐s♠ ❆ ❣✐❢t

• ■ ❛s ❛ ❣✐❢t

◮ ❉♦❡s P❂◆P❄ ❙✳❙♠❛❧❡✿ ■ s♦♠❡t✐♠❡s ❝♦♥s✐❞❡r t❤✐s ♣r♦❜❧❡♠ ❛s ❛

❣✐❢t t♦ ♠❛t❤❡♠❛t✐❝s ❢r♦♠ ❝♦♠♣✉t❡r s❝✐❡♥❝❡✳

◮ ❉✐♦♣❤❛♥t(Z/2)✿ ●✐✈❡♥ ✭❛s ✐♥♣✉t✮ k ♣♦❧②♥♦♠✐❛❧s ✐♥ n ✈❛r✐❛❜❧❡s

✇✐t❤ ❝♦❡✣❝✐❡♥ts ✐♥ Z/2✳ ❉❡❝✐❞❡ ✐❢ t❤❡r❡ ✐s ❛ ❝♦♠♠♦♥ ③❡r♦ x ∈ (Z/2)n❄

◮ ❈♦♥❥❡❝t✉r❡✳ ❚❤❡r❡ ✐s ♥♦ ♣♦❧②♥♦♠✐❛❧ t✐♠❡ ❛❧❣♦r✐t❤♠ ♦✈❡r Z/2

❞❡❝✐❞✐♥❣ t❤✐s ♣r♦❜❧❡♠✳ ❚❤✐s ✐s ❛ r❡❢♦r♠✉❧❛t✐♦♥ ♦❢ t❤❡ ❝❧❛ss✐❝ ❝♦♥❥❡❝t✉r❡ P = NP✳ ◗✉❡st✐♦♥s✳ ❲❤❛t ✐s t❤❡ r❡❞✉❝t✐♦♥ r❡❧❛t✐♦♥ ❜❡t✇❡❡♥ ●■ ❛♥❞ ❄ ❙♣❡❝✐✜❝❛❧❧②✱ ❝❛♥ ❜❡ ♣♦❧②♥♦♠✐❛❧❧② r❡❞✉❝❡❞ t♦ ❄ ■s ♣♦❧②♥♦♠✐❛❧❧② r❡❞✉❝✐❜❧❡ t♦ ❄

❙❡❛r❝❤✐♥❣ ❢♦r ❛ ●r❛♣❤ ■s♦♠♦r♣❤✐s♠ ❆ ❣✐❢t

• ■ ❛s ❛ ❣✐❢t

◮ ❉♦❡s P❂◆P❄ ❙✳❙♠❛❧❡✿ ■ s♦♠❡t✐♠❡s ❝♦♥s✐❞❡r t❤✐s ♣r♦❜❧❡♠ ❛s ❛

❣✐❢t t♦ ♠❛t❤❡♠❛t✐❝s ❢r♦♠ ❝♦♠♣✉t❡r s❝✐❡♥❝❡✳

◮ ❉✐♦♣❤❛♥t(Z/2)✿ ●✐✈❡♥ ✭❛s ✐♥♣✉t✮ k ♣♦❧②♥♦♠✐❛❧s ✐♥ n ✈❛r✐❛❜❧❡s

✇✐t❤ ❝♦❡✣❝✐❡♥ts ✐♥ Z/2✳ ❉❡❝✐❞❡ ✐❢ t❤❡r❡ ✐s ❛ ❝♦♠♠♦♥ ③❡r♦ x ∈ (Z/2)n❄

◮ ❈♦♥❥❡❝t✉r❡✳ ❚❤❡r❡ ✐s ♥♦ ♣♦❧②♥♦♠✐❛❧ t✐♠❡ ❛❧❣♦r✐t❤♠ ♦✈❡r Z/2

❞❡❝✐❞✐♥❣ t❤✐s ♣r♦❜❧❡♠✳ ❚❤✐s ✐s ❛ r❡❢♦r♠✉❧❛t✐♦♥ ♦❢ t❤❡ ❝❧❛ss✐❝ ❝♦♥❥❡❝t✉r❡ P = NP✳

◮ ◗✉❡st✐♦♥s✳ ❲❤❛t ✐s t❤❡ r❡❞✉❝t✐♦♥ r❡❧❛t✐♦♥ ❜❡t✇❡❡♥ ●■ ❛♥❞

Diophant(Z/2)❄ ❙♣❡❝✐✜❝❛❧❧②✱ ❝❛♥ GI ❜❡ ♣♦❧②♥♦♠✐❛❧❧② r❡❞✉❝❡❞ t♦ Diophant(Z/2)❄ ■s SeachDiophant(Z/2) ♣♦❧②♥♦♠✐❛❧❧② r❡❞✉❝✐❜❧❡ t♦ Diophant(Z/2)❄

❙❡❛r❝❤✐♥❣ ❢♦r ❛ ●r❛♣❤ ■s♦♠♦r♣❤✐s♠ ❆ ❣✐❢t

• ■ ❛s ❛ ❣✐❢t

◮ ❉♦❡s P❂◆P❄ ❙✳❙♠❛❧❡✿ ■ s♦♠❡t✐♠❡s ❝♦♥s✐❞❡r t❤✐s ♣r♦❜❧❡♠ ❛s ❛

❣✐❢t t♦ ♠❛t❤❡♠❛t✐❝s ❢r♦♠ ❝♦♠♣✉t❡r s❝✐❡♥❝❡✳

◮ ❉✐♦♣❤❛♥t(Z/2)✿ ●✐✈❡♥ ✭❛s ✐♥♣✉t✮ k ♣♦❧②♥♦♠✐❛❧s ✐♥ n ✈❛r✐❛❜❧❡s

✇✐t❤ ❝♦❡✣❝✐❡♥ts ✐♥ Z/2✳ ❉❡❝✐❞❡ ✐❢ t❤❡r❡ ✐s ❛ ❝♦♠♠♦♥ ③❡r♦ x ∈ (Z/2)n❄

◮ ❈♦♥❥❡❝t✉r❡✳ ❚❤❡r❡ ✐s ♥♦ ♣♦❧②♥♦♠✐❛❧ t✐♠❡ ❛❧❣♦r✐t❤♠ ♦✈❡r Z/2

❞❡❝✐❞✐♥❣ t❤✐s ♣r♦❜❧❡♠✳ ❚❤✐s ✐s ❛ r❡❢♦r♠✉❧❛t✐♦♥ ♦❢ t❤❡ ❝❧❛ss✐❝ ❝♦♥❥❡❝t✉r❡ P = NP✳

◮ ◗✉❡st✐♦♥s✳ ❲❤❛t ✐s t❤❡ r❡❞✉❝t✐♦♥ r❡❧❛t✐♦♥ ❜❡t✇❡❡♥ ●■ ❛♥❞

Diophant(Z/2)❄ ❙♣❡❝✐✜❝❛❧❧②✱ ❝❛♥ GI ❜❡ ♣♦❧②♥♦♠✐❛❧❧② r❡❞✉❝❡❞ t♦ Diophant(Z/2)❄ ■s SeachDiophant(Z/2) ♣♦❧②♥♦♠✐❛❧❧② r❡❞✉❝✐❜❧❡ t♦ Diophant(Z/2)❄

❙❡❛r❝❤✐♥❣ ❢♦r ❛ ●r❛♣❤ ■s♦♠♦r♣❤✐s♠ ❆ ❣✐❢t

❙❡❛r❝❤✐♥❣ ❢♦r ❛ ●r❛♣❤ ■s♦♠♦r♣❤✐s♠ ❙♦❧✈✐♥❣ ●■ ❛✳❛✳s✉r❡❧②

❆❧❣♦r✐t❤♠ ▲❆❇❊▲ ✭❇❛❜❛✐✱ ❊r❞♦s ❛♥❞ ❙❡❧❦♦✇ ✭✶✾✽✵✮✮

❈❛♥♦♥✐❝❛❧ ❧❛❜❡❧❧✐♥❣ ❛ ❣r❛♣❤ ✳ ❙t❡♣ ✵✿ ■♥♣✉t ❣r❛♣❤ ❛♥❞ ♣❛r❛♠❡t❡r ✳ ❙t❡♣ ✶✿ ❘❡✲❧❛❜❡❧ t❤❡ ✈❡rt✐❝❡s ♦❢ s♦ t❤❛t t❤❡② s❛t✐s❢② ✿ ■❢ t❤❡r❡ ❡①✐sts s✉❝❤ t❤❛t ✱ t❤❡♥ ❋❆■▲✳ ❙t❡♣ ✷✿ ❋♦r ❧❡t ❘❡✲❧❛❜❡❧ ✈❡rt✐❝❡s s♦ t❤❛t t❤❡s❡ s❡ts s❛t✐s❢② ✇❤❡r❡ ❞❡♥♦t❡s ❧❡①✐❝♦❣r❛♣❤✐❝ ♦r❞❡r✳ ■❢ t❤❡r❡ ❡①✐sts s✉❝❤ t❤❛t t❤❡♥ ❋❆■▲✳

❙❡❛r❝❤✐♥❣ ❢♦r ❛ ●r❛♣❤ ■s♦♠♦r♣❤✐s♠ ❙♦❧✈✐♥❣ ●■ ❛✳❛✳s✉r❡❧②

❆❧❣♦r✐t❤♠ ▲❆❇❊▲ ✭❇❛❜❛✐✱ ❊r❞♦s ❛♥❞ ❙❡❧❦♦✇ ✭✶✾✽✵✮✮

◮ ❈❛♥♦♥✐❝❛❧ ❧❛❜❡❧❧✐♥❣ ❛ ❣r❛♣❤ G✳

❙t❡♣ ✵✿ ■♥♣✉t ❣r❛♣❤ ❛♥❞ ♣❛r❛♠❡t❡r ✳ ❙t❡♣ ✶✿ ❘❡✲❧❛❜❡❧ t❤❡ ✈❡rt✐❝❡s ♦❢ s♦ t❤❛t t❤❡② s❛t✐s❢② ✿ ■❢ t❤❡r❡ ❡①✐sts s✉❝❤ t❤❛t ✱ t❤❡♥ ❋❆■▲✳ ❙t❡♣ ✷✿ ❋♦r ❧❡t ❘❡✲❧❛❜❡❧ ✈❡rt✐❝❡s s♦ t❤❛t t❤❡s❡ s❡ts s❛t✐s❢② ✇❤❡r❡ ❞❡♥♦t❡s ❧❡①✐❝♦❣r❛♣❤✐❝ ♦r❞❡r✳ ■❢ t❤❡r❡ ❡①✐sts s✉❝❤ t❤❛t t❤❡♥ ❋❆■▲✳

❙❡❛r❝❤✐♥❣ ❢♦r ❛ ●r❛♣❤ ■s♦♠♦r♣❤✐s♠ ❙♦❧✈✐♥❣ ●■ ❛✳❛✳s✉r❡❧②

❆❧❣♦r✐t❤♠ ▲❆❇❊▲ ✭❇❛❜❛✐✱ ❊r❞♦s ❛♥❞ ❙❡❧❦♦✇ ✭✶✾✽✵✮✮

◮ ❈❛♥♦♥✐❝❛❧ ❧❛❜❡❧❧✐♥❣ ❛ ❣r❛♣❤ G✳ ◮ ❙t❡♣ ✵✿ ■♥♣✉t ❣r❛♣❤ G ❛♥❞ ♣❛r❛♠❡t❡r L = 3 log2 n✳

❙t❡♣ ✶✿ ❘❡✲❧❛❜❡❧ t❤❡ ✈❡rt✐❝❡s ♦❢ s♦ t❤❛t t❤❡② s❛t✐s❢② ✿ ■❢ t❤❡r❡ ❡①✐sts s✉❝❤ t❤❛t ✱ t❤❡♥ ❋❆■▲✳ ❙t❡♣ ✷✿ ❋♦r ❧❡t ❘❡✲❧❛❜❡❧ ✈❡rt✐❝❡s s♦ t❤❛t t❤❡s❡ s❡ts s❛t✐s❢② ✇❤❡r❡ ❞❡♥♦t❡s ❧❡①✐❝♦❣r❛♣❤✐❝ ♦r❞❡r✳ ■❢ t❤❡r❡ ❡①✐sts s✉❝❤ t❤❛t t❤❡♥ ❋❆■▲✳

❙❡❛r❝❤✐♥❣ ❢♦r ❛ ●r❛♣❤ ■s♦♠♦r♣❤✐s♠ ❙♦❧✈✐♥❣ ●■ ❛✳❛✳s✉r❡❧②

❆❧❣♦r✐t❤♠ ▲❆❇❊▲ ✭❇❛❜❛✐✱ ❊r❞♦s ❛♥❞ ❙❡❧❦♦✇ ✭✶✾✽✵✮✮

◮ ❈❛♥♦♥✐❝❛❧ ❧❛❜❡❧❧✐♥❣ ❛ ❣r❛♣❤ G✳ ◮ ❙t❡♣ ✵✿ ■♥♣✉t ❣r❛♣❤ G ❛♥❞ ♣❛r❛♠❡t❡r L = 3 log2 n✳ ◮ ❙t❡♣ ✶✿ ❘❡✲❧❛❜❡❧ t❤❡ ✈❡rt✐❝❡s ♦❢ G s♦ t❤❛t t❤❡② s❛t✐s❢②

dG(v1) ≥ dG(v2) ≥ · · · ≥ dG(vn) ✿ ■❢ t❤❡r❡ ❡①✐sts s✉❝❤ t❤❛t ✱ t❤❡♥ ❋❆■▲✳ ❙t❡♣ ✷✿ ❋♦r ❧❡t ❘❡✲❧❛❜❡❧ ✈❡rt✐❝❡s s♦ t❤❛t t❤❡s❡ s❡ts s❛t✐s❢② ✇❤❡r❡ ❞❡♥♦t❡s ❧❡①✐❝♦❣r❛♣❤✐❝ ♦r❞❡r✳ ■❢ t❤❡r❡ ❡①✐sts s✉❝❤ t❤❛t t❤❡♥ ❋❆■▲✳

❙❡❛r❝❤✐♥❣ ❢♦r ❛ ●r❛♣❤ ■s♦♠♦r♣❤✐s♠ ❙♦❧✈✐♥❣ ●■ ❛✳❛✳s✉r❡❧②

❆❧❣♦r✐t❤♠ ▲❆❇❊▲ ✭❇❛❜❛✐✱ ❊r❞♦s ❛♥❞ ❙❡❧❦♦✇ ✭✶✾✽✵✮✮

◮ ❈❛♥♦♥✐❝❛❧ ❧❛❜❡❧❧✐♥❣ ❛ ❣r❛♣❤ G✳ ◮ ❙t❡♣ ✵✿ ■♥♣✉t ❣r❛♣❤ G ❛♥❞ ♣❛r❛♠❡t❡r L = 3 log2 n✳ ◮ ❙t❡♣ ✶✿ ❘❡✲❧❛❜❡❧ t❤❡ ✈❡rt✐❝❡s ♦❢ G s♦ t❤❛t t❤❡② s❛t✐s❢②

dG(v1) ≥ dG(v2) ≥ · · · ≥ dG(vn) ✿

◮ ■❢ t❤❡r❡ ❡①✐sts i < L s✉❝❤ t❤❛t dG(vi) = dG(vi+1)✱ t❤❡♥ ❋❆■▲✳

❙t❡♣ ✷✿ ❋♦r ❧❡t ❘❡✲❧❛❜❡❧ ✈❡rt✐❝❡s s♦ t❤❛t t❤❡s❡ s❡ts s❛t✐s❢② ✇❤❡r❡ ❞❡♥♦t❡s ❧❡①✐❝♦❣r❛♣❤✐❝ ♦r❞❡r✳ ■❢ t❤❡r❡ ❡①✐sts s✉❝❤ t❤❛t t❤❡♥ ❋❆■▲✳

❙❡❛r❝❤✐♥❣ ❢♦r ❛ ●r❛♣❤ ■s♦♠♦r♣❤✐s♠ ❙♦❧✈✐♥❣ ●■ ❛✳❛✳s✉r❡❧②

❆❧❣♦r✐t❤♠ ▲❆❇❊▲ ✭❇❛❜❛✐✱ ❊r❞♦s ❛♥❞ ❙❡❧❦♦✇ ✭✶✾✽✵✮✮

◮ ❈❛♥♦♥✐❝❛❧ ❧❛❜❡❧❧✐♥❣ ❛ ❣r❛♣❤ G✳ ◮ ❙t❡♣ ✵✿ ■♥♣✉t ❣r❛♣❤ G ❛♥❞ ♣❛r❛♠❡t❡r L = 3 log2 n✳ ◮ ❙t❡♣ ✶✿ ❘❡✲❧❛❜❡❧ t❤❡ ✈❡rt✐❝❡s ♦❢ G s♦ t❤❛t t❤❡② s❛t✐s❢②

dG(v1) ≥ dG(v2) ≥ · · · ≥ dG(vn) ✿

◮ ■❢ t❤❡r❡ ❡①✐sts i < L s✉❝❤ t❤❛t dG(vi) = dG(vi+1)✱ t❤❡♥ ❋❆■▲✳ ◮ ❙t❡♣ ✷✿ ❋♦r i > L ❧❡t

Xi = {j ∈ {1, 2, . . . , L} : {vi, vj} ∈ EG. ❘❡✲❧❛❜❡❧ ✈❡rt✐❝❡s vL+1, vL+2, ..., vn s♦ t❤❛t t❤❡s❡ s❡ts s❛t✐s❢② XL+1 ≻ XL+2 ≻ · · · ≻ Xn, ✇❤❡r❡ ≻ ❞❡♥♦t❡s ❧❡①✐❝♦❣r❛♣❤✐❝ ♦r❞❡r✳ ■❢ t❤❡r❡ ❡①✐sts i < n s✉❝❤ t❤❛t Xi = Xi+1 t❤❡♥ ❋❆■▲✳

❙❡❛r❝❤✐♥❣ ❢♦r ❛ ●r❛♣❤ ■s♦♠♦r♣❤✐s♠ ❙♦❧✈✐♥❣ ●■ ❛✳❛✳s✉r❡❧②

❆❧❣♦r✐t❤♠ ▲❆❇❊▲ ✭❇❛❜❛✐✱ ❊r❞♦s ❛♥❞ ❙❡❧❦♦✇ ✭✶✾✽✵✮✮

◮ ❈❛♥♦♥✐❝❛❧ ❧❛❜❡❧❧✐♥❣ ❛ ❣r❛♣❤ G✳ ◮ ❙t❡♣ ✵✿ ■♥♣✉t ❣r❛♣❤ G ❛♥❞ ♣❛r❛♠❡t❡r L = 3 log2 n✳ ◮ ❙t❡♣ ✶✿ ❘❡✲❧❛❜❡❧ t❤❡ ✈❡rt✐❝❡s ♦❢ G s♦ t❤❛t t❤❡② s❛t✐s❢②

dG(v1) ≥ dG(v2) ≥ · · · ≥ dG(vn) ✿

◮ ■❢ t❤❡r❡ ❡①✐sts i < L s✉❝❤ t❤❛t dG(vi) = dG(vi+1)✱ t❤❡♥ ❋❆■▲✳ ◮ ❙t❡♣ ✷✿ ❋♦r i > L ❧❡t

Xi = {j ∈ {1, 2, . . . , L} : {vi, vj} ∈ EG. ❘❡✲❧❛❜❡❧ ✈❡rt✐❝❡s vL+1, vL+2, ..., vn s♦ t❤❛t t❤❡s❡ s❡ts s❛t✐s❢② XL+1 ≻ XL+2 ≻ · · · ≻ Xn, ✇❤❡r❡ ≻ ❞❡♥♦t❡s ❧❡①✐❝♦❣r❛♣❤✐❝ ♦r❞❡r✳ ■❢ t❤❡r❡ ❡①✐sts i < n s✉❝❤ t❤❛t Xi = Xi+1 t❤❡♥ ❋❆■▲✳

❙❡❛r❝❤✐♥❣ ❢♦r ❛ ●r❛♣❤ ■s♦♠♦r♣❤✐s♠ ❙♦❧✈✐♥❣ ●■ ❛✳❛✳s✉r❡❧②

❘✳▲✐♣t♦♥ ❛♥❞ ❘✳❑❛r♣

◮

❘✳ ▲✐♣t♦♥✱ ❚❤❡ ❜❡❛❝♦♥ s❡t ❛♣♣r♦❛❝❤ t♦ ❣r❛♣❤ ✐s♦♠♦r♣❤✐s♠✱ ❘❡s❡❛r❝❤ ❘❡♣♦rt ✶✸✺✱ ❨❛❧❡ ❯♥✐✈❡rs✐t②✱ ✶✾✼✽✳ ❘✐❝❤❛r❞ ▼✳ ❑❛r♣✱ Pr♦❜❛❜✐❧✐st✐❝ ❛♥❛❧②s✐s ♦❢ ❛ ❝❛♥♦♥✐❝❛❧ ♥✉♠❜❡r✐♥❣ ❛❧❣♦r✐t❤♠ ❢♦r ❣r❛♣❤s ✭✶✾✼✾✮✳

❙❡❛r❝❤✐♥❣ ❢♦r ❛ ●r❛♣❤ ■s♦♠♦r♣❤✐s♠ ❙♦❧✈✐♥❣ ●■ ❛✳❛✳s✉r❡❧②

❘✳▲✐♣t♦♥ ❛♥❞ ❘✳❑❛r♣

◮ ◮ ❘✳ ▲✐♣t♦♥✱ ❚❤❡ ❜❡❛❝♦♥ s❡t ❛♣♣r♦❛❝❤ t♦ ❣r❛♣❤ ✐s♦♠♦r♣❤✐s♠✱

❘❡s❡❛r❝❤ ❘❡♣♦rt ✶✸✺✱ ❨❛❧❡ ❯♥✐✈❡rs✐t②✱ ✶✾✼✽✳ ❘✐❝❤❛r❞ ▼✳ ❑❛r♣✱ Pr♦❜❛❜✐❧✐st✐❝ ❛♥❛❧②s✐s ♦❢ ❛ ❝❛♥♦♥✐❝❛❧ ♥✉♠❜❡r✐♥❣ ❛❧❣♦r✐t❤♠ ❢♦r ❣r❛♣❤s ✭✶✾✼✾✮✳

❙❡❛r❝❤✐♥❣ ❢♦r ❛ ●r❛♣❤ ■s♦♠♦r♣❤✐s♠ ❙♦❧✈✐♥❣ ●■ ❛✳❛✳s✉r❡❧②

❘✳▲✐♣t♦♥ ❛♥❞ ❘✳❑❛r♣

◮ ◮ ❘✳ ▲✐♣t♦♥✱ ❚❤❡ ❜❡❛❝♦♥ s❡t ❛♣♣r♦❛❝❤ t♦ ❣r❛♣❤ ✐s♦♠♦r♣❤✐s♠✱

❘❡s❡❛r❝❤ ❘❡♣♦rt ✶✸✺✱ ❨❛❧❡ ❯♥✐✈❡rs✐t②✱ ✶✾✼✽✳

◮

❘✐❝❤❛r❞ ▼✳ ❑❛r♣✱ Pr♦❜❛❜✐❧✐st✐❝ ❛♥❛❧②s✐s ♦❢ ❛ ❝❛♥♦♥✐❝❛❧ ♥✉♠❜❡r✐♥❣ ❛❧❣♦r✐t❤♠ ❢♦r ❣r❛♣❤s ✭✶✾✼✾✮✳

❙❡❛r❝❤✐♥❣ ❢♦r ❛ ●r❛♣❤ ■s♦♠♦r♣❤✐s♠ ❙♦❧✈✐♥❣ ●■ ❛✳❛✳s✉r❡❧②

❘✳▲✐♣t♦♥ ❛♥❞ ❘✳❑❛r♣

◮ ◮ ❘✳ ▲✐♣t♦♥✱ ❚❤❡ ❜❡❛❝♦♥ s❡t ❛♣♣r♦❛❝❤ t♦ ❣r❛♣❤ ✐s♦♠♦r♣❤✐s♠✱

❘❡s❡❛r❝❤ ❘❡♣♦rt ✶✸✺✱ ❨❛❧❡ ❯♥✐✈❡rs✐t②✱ ✶✾✼✽✳

◮ ◮ ❘✐❝❤❛r❞ ▼✳ ❑❛r♣✱ Pr♦❜❛❜✐❧✐st✐❝ ❛♥❛❧②s✐s ♦❢ ❛ ❝❛♥♦♥✐❝❛❧

♥✉♠❜❡r✐♥❣ ❛❧❣♦r✐t❤♠ ❢♦r ❣r❛♣❤s ✭✶✾✼✾✮✳

❙❡❛r❝❤✐♥❣ ❢♦r ❛ ●r❛♣❤ ■s♦♠♦r♣❤✐s♠ ❙♦❧✈✐♥❣ ●■ ❛✳❛✳s✉r❡❧②

❘✳▲✐♣t♦♥ ❛♥❞ ❘✳❑❛r♣

◮ ◮ ❘✳ ▲✐♣t♦♥✱ ❚❤❡ ❜❡❛❝♦♥ s❡t ❛♣♣r♦❛❝❤ t♦ ❣r❛♣❤ ✐s♦♠♦r♣❤✐s♠✱

❘❡s❡❛r❝❤ ❘❡♣♦rt ✶✸✺✱ ❨❛❧❡ ❯♥✐✈❡rs✐t②✱ ✶✾✼✽✳

◮ ◮ ❘✐❝❤❛r❞ ▼✳ ❑❛r♣✱ Pr♦❜❛❜✐❧✐st✐❝ ❛♥❛❧②s✐s ♦❢ ❛ ❝❛♥♦♥✐❝❛❧

♥✉♠❜❡r✐♥❣ ❛❧❣♦r✐t❤♠ ❢♦r ❣r❛♣❤s ✭✶✾✼✾✮✳

❙❡❛r❝❤✐♥❣ ❢♦r ❛ ●r❛♣❤ ■s♦♠♦r♣❤✐s♠ ❙♦❧✈✐♥❣ ●■ ❛✳❛✳s✉r❡❧②

❈♦♠♣❛r✐s♦♥ ♦❢ ✸ ❛❧❣♦r✐t❤♠s

❙❡❛r❝❤✐♥❣ ❢♦r ❛ ●r❛♣❤ ■s♦♠♦r♣❤✐s♠ ❊✲t❡st

❊❞❣❡ ❣r❛♣❤ ✐s♦♠♦r♣❤✐s♠ t❡st ✭❊✲t❡st✮✳

◮ ❚❤❡ ✐♥♣✉t s❡t I = ∪n(Gn × Gn)✳

❊✲t❡st✿ ●✐✈❡♥

✶✳ ❈♦♠♣❛r❡ ✱ ✷✳ ■❢ t❤❡ t❡st s❛②s t❤❛t ❛r❡ ♥♦♥✲✐s♦♠♦r♣❤✐❝✳ ✸✳ ■❢ t❤❡ t❡st s❛②s ✧■ ❞♦♥✬t ❦♥♦✇✧✳

❚❤❡ ❢❛✐❧✉r❡ s❡t ♦❢ ❊✲t❡st ✐s t❤❡ s❡t ♦❢ ♣❛✐rs ❚♦ ♠❡❛s✉r❡ t❤❡ s✐③❡ ♦❢ t❤❡ ❢❛✐❧✉r❡ s❡t ♦♥❡ ♥❡❡❞s ❛♥ ❡♥s❡♠❜❧❡ ♦❢ ❞✐str✐❜✉t✐♦♥s ♦♥ ❋♦r ✐♥st❛♥❝❡ t❤❡ ❜✐♥♦♠✐❛❧ ❞✐str✐❜✉t✐♦♥ ✇✐t❤ ♣❛r❛♠❡t❡rs ✇❤❡r❡

❙❡❛r❝❤✐♥❣ ❢♦r ❛ ●r❛♣❤ ■s♦♠♦r♣❤✐s♠ ❊✲t❡st

❊❞❣❡ ❣r❛♣❤ ✐s♦♠♦r♣❤✐s♠ t❡st ✭❊✲t❡st✮✳

◮ ❚❤❡ ✐♥♣✉t s❡t I = ∪n(Gn × Gn)✳ ◮ ❊✲t❡st✿ ●✐✈❡♥ G1, G2

✶✳ ❈♦♠♣❛r❡ |EG1| , |EG2|✱ ✷✳ ■❢ |EG1| = |EG2| t❤❡ t❡st s❛②s t❤❛t G1, G2 ❛r❡ ♥♦♥✲✐s♦♠♦r♣❤✐❝✳ ✸✳ ■❢ |EG1| = |EG2| t❤❡ t❡st s❛②s ✧■ ❞♦♥✬t ❦♥♦✇✧✳

❚❤❡ ❢❛✐❧✉r❡ s❡t ♦❢ ❊✲t❡st ✐s t❤❡ s❡t ♦❢ ♣❛✐rs ❚♦ ♠❡❛s✉r❡ t❤❡ s✐③❡ ♦❢ t❤❡ ❢❛✐❧✉r❡ s❡t ♦♥❡ ♥❡❡❞s ❛♥ ❡♥s❡♠❜❧❡ ♦❢ ❞✐str✐❜✉t✐♦♥s ♦♥ ❋♦r ✐♥st❛♥❝❡ t❤❡ ❜✐♥♦♠✐❛❧ ❞✐str✐❜✉t✐♦♥ ✇✐t❤ ♣❛r❛♠❡t❡rs ✇❤❡r❡

❙❡❛r❝❤✐♥❣ ❢♦r ❛ ●r❛♣❤ ■s♦♠♦r♣❤✐s♠ ❊✲t❡st

❊❞❣❡ ❣r❛♣❤ ✐s♦♠♦r♣❤✐s♠ t❡st ✭❊✲t❡st✮✳

◮ ❚❤❡ ✐♥♣✉t s❡t I = ∪n(Gn × Gn)✳ ◮ ❊✲t❡st✿ ●✐✈❡♥ G1, G2

✶✳ ❈♦♠♣❛r❡ |EG1| , |EG2|✱ ✷✳ ■❢ |EG1| = |EG2| t❤❡ t❡st s❛②s t❤❛t G1, G2 ❛r❡ ♥♦♥✲✐s♦♠♦r♣❤✐❝✳ ✸✳ ■❢ |EG1| = |EG2| t❤❡ t❡st s❛②s ✧■ ❞♦♥✬t ❦♥♦✇✧✳

◮ ❚❤❡ ❢❛✐❧✉r❡ s❡t ♦❢ ❊✲t❡st ✐s t❤❡ s❡t ♦❢ ♣❛✐rs

Fn = {(G1, G2) : G1, G2 ∈ G (n) , |EG1| = |EG2|} . ❚♦ ♠❡❛s✉r❡ t❤❡ s✐③❡ ♦❢ t❤❡ ❢❛✐❧✉r❡ s❡t ♦♥❡ ♥❡❡❞s ❛♥ ❡♥s❡♠❜❧❡ ♦❢ ❞✐str✐❜✉t✐♦♥s ♦♥ ❋♦r ✐♥st❛♥❝❡ t❤❡ ❜✐♥♦♠✐❛❧ ❞✐str✐❜✉t✐♦♥ ✇✐t❤ ♣❛r❛♠❡t❡rs ✇❤❡r❡

❙❡❛r❝❤✐♥❣ ❢♦r ❛ ●r❛♣❤ ■s♦♠♦r♣❤✐s♠ ❊✲t❡st

❊❞❣❡ ❣r❛♣❤ ✐s♦♠♦r♣❤✐s♠ t❡st ✭❊✲t❡st✮✳

◮ ❚❤❡ ✐♥♣✉t s❡t I = ∪n(Gn × Gn)✳ ◮ ❊✲t❡st✿ ●✐✈❡♥ G1, G2

✶✳ ❈♦♠♣❛r❡ |EG1| , |EG2|✱ ✷✳ ■❢ |EG1| = |EG2| t❤❡ t❡st s❛②s t❤❛t G1, G2 ❛r❡ ♥♦♥✲✐s♦♠♦r♣❤✐❝✳ ✸✳ ■❢ |EG1| = |EG2| t❤❡ t❡st s❛②s ✧■ ❞♦♥✬t ❦♥♦✇✧✳

◮ ❚❤❡ ❢❛✐❧✉r❡ s❡t ♦❢ ❊✲t❡st ✐s t❤❡ s❡t ♦❢ ♣❛✐rs

Fn = {(G1, G2) : G1, G2 ∈ G (n) , |EG1| = |EG2|} .

◮ ❚♦ ♠❡❛s✉r❡ t❤❡ s✐③❡ ♦❢ t❤❡ ❢❛✐❧✉r❡ s❡t ♦♥❡ ♥❡❡❞s ❛♥ ❡♥s❡♠❜❧❡

♦❢ ❞✐str✐❜✉t✐♦♥s ♦♥ G(n). ❋♦r ✐♥st❛♥❝❡ t❤❡ ❜✐♥♦♠✐❛❧ ❞✐str✐❜✉t✐♦♥ Pn ✇✐t❤ ♣❛r❛♠❡t❡rs N = n(n−1)

2

, p. ✇❤❡r❡

❙❡❛r❝❤✐♥❣ ❢♦r ❛ ●r❛♣❤ ■s♦♠♦r♣❤✐s♠ ❊✲t❡st

❊❞❣❡ ❣r❛♣❤ ✐s♦♠♦r♣❤✐s♠ t❡st ✭❊✲t❡st✮✳

◮ ❚❤❡ ✐♥♣✉t s❡t I = ∪n(Gn × Gn)✳ ◮ ❊✲t❡st✿ ●✐✈❡♥ G1, G2

✶✳ ❈♦♠♣❛r❡ |EG1| , |EG2|✱ ✷✳ ■❢ |EG1| = |EG2| t❤❡ t❡st s❛②s t❤❛t G1, G2 ❛r❡ ♥♦♥✲✐s♦♠♦r♣❤✐❝✳ ✸✳ ■❢ |EG1| = |EG2| t❤❡ t❡st s❛②s ✧■ ❞♦♥✬t ❦♥♦✇✧✳

◮ ❚❤❡ ❢❛✐❧✉r❡ s❡t ♦❢ ❊✲t❡st ✐s t❤❡ s❡t ♦❢ ♣❛✐rs

Fn = {(G1, G2) : G1, G2 ∈ G (n) , |EG1| = |EG2|} .

◮ ❚♦ ♠❡❛s✉r❡ t❤❡ s✐③❡ ♦❢ t❤❡ ❢❛✐❧✉r❡ s❡t ♦♥❡ ♥❡❡❞s ❛♥ ❡♥s❡♠❜❧❡

♦❢ ❞✐str✐❜✉t✐♦♥s ♦♥ G(n). ❋♦r ✐♥st❛♥❝❡ t❤❡ ❜✐♥♦♠✐❛❧ ❞✐str✐❜✉t✐♦♥ Pn ✇✐t❤ ♣❛r❛♠❡t❡rs N = n(n−1)

2

, p.

◮ Pn(G) = pkqN−k, ✇❤❡r❡ k = |E|, q = 1 − p, 0 < p < 1.

❙❡❛r❝❤✐♥❣ ❢♦r ❛ ●r❛♣❤ ■s♦♠♦r♣❤✐s♠ ❊✲t❡st

❊❞❣❡ ❣r❛♣❤ ✐s♦♠♦r♣❤✐s♠ t❡st ✭❊✲t❡st✮✳

◮ ❚❤❡ ✐♥♣✉t s❡t I = ∪n(Gn × Gn)✳ ◮ ❊✲t❡st✿ ●✐✈❡♥ G1, G2

✶✳ ❈♦♠♣❛r❡ |EG1| , |EG2|✱ ✷✳ ■❢ |EG1| = |EG2| t❤❡ t❡st s❛②s t❤❛t G1, G2 ❛r❡ ♥♦♥✲✐s♦♠♦r♣❤✐❝✳ ✸✳ ■❢ |EG1| = |EG2| t❤❡ t❡st s❛②s ✧■ ❞♦♥✬t ❦♥♦✇✧✳

◮ ❚❤❡ ❢❛✐❧✉r❡ s❡t ♦❢ ❊✲t❡st ✐s t❤❡ s❡t ♦❢ ♣❛✐rs

Fn = {(G1, G2) : G1, G2 ∈ G (n) , |EG1| = |EG2|} .

◮ ❚♦ ♠❡❛s✉r❡ t❤❡ s✐③❡ ♦❢ t❤❡ ❢❛✐❧✉r❡ s❡t ♦♥❡ ♥❡❡❞s ❛♥ ❡♥s❡♠❜❧❡

♦❢ ❞✐str✐❜✉t✐♦♥s ♦♥ G(n). ❋♦r ✐♥st❛♥❝❡ t❤❡ ❜✐♥♦♠✐❛❧ ❞✐str✐❜✉t✐♦♥ Pn ✇✐t❤ ♣❛r❛♠❡t❡rs N = n(n−1)

2

, p.

◮ Pn(G) = pkqN−k, ✇❤❡r❡ k = |E|, q = 1 − p, 0 < p < 1.

❙❡❛r❝❤✐♥❣ ❢♦r ❛ ●r❛♣❤ ■s♦♠♦r♣❤✐s♠ ❊✲t❡st

❋❛✐❧✉r❡ ♣r♦❜❛❜✐❧✐t②

◮ ❚❤❡♦r❡♠ ✭●✳❆✳◆♦s❦♦✈✲❆✳◆✳❘②❜❛❧♦✈✱✷✵✶✹✮

▲❡t Fn ❜❡ t❤❡ ❢❛✐❧✉r❡ s❡t ♦❢ t❤❡ ❊✲t❡st ♦♥ t❤❡ s❡t ♦❢ ✐♥♣✉ts G (n, p) × G (n, p) , n ∈ N✱ 0 < p ≤ 1

2✳

✶✳ ■❢ t❤❡♥ ❤❛♣♣❡♥s ❛✳❛✳s✳✱ ✐✳❡✳ t❤❡ ❢❛✐❧✉r❡ ♣r♦❜❛❜✐❧✐t② t❡♥❞s t♦ ✶ ❛s t❡♥❞s t♦ ✐♥✜♥✐t②✱ ✷✳ ■❢ ✱ ✱ t❤❡♥ ✱ ✇❤❡r❡ ✐s t❤❡ P♦✐ss♦♥ ❞✐str✐❜✉t✐♦♥ ✈❡❝t♦r ❛♥❞ ✸✳ ■❢ t❤❡♥ ✐s ♥❡❣❧✐❣✐❜❧❡ ✭✐✳❡✳ ❛s ✮ ❛♥❞✱ ♠♦r❡♦✈❡r✱ ❢♦r ✳ ✹✳ ■❢✱ ✐♥ ❝❛s❡ ✸✱ ✐s ❝♦♥st❛♥t✱ t❤❡♥ ✐s ❛s②♠♣t♦t✐❝ t♦ ✳

❙❡❛r❝❤✐♥❣ ❢♦r ❛ ●r❛♣❤ ■s♦♠♦r♣❤✐s♠ ❊✲t❡st

❋❛✐❧✉r❡ ♣r♦❜❛❜✐❧✐t②

◮ ❚❤❡♦r❡♠ ✭●✳❆✳◆♦s❦♦✈✲❆✳◆✳❘②❜❛❧♦✈✱✷✵✶✹✮

▲❡t Fn ❜❡ t❤❡ ❢❛✐❧✉r❡ s❡t ♦❢ t❤❡ ❊✲t❡st ♦♥ t❤❡ s❡t ♦❢ ✐♥♣✉ts G (n, p) × G (n, p) , n ∈ N✱ 0 < p ≤ 1

2✳

✶✳ ■❢ limn→∞ np = 0 , t❤❡♥ F = ∪Fn ❤❛♣♣❡♥s ❛✳❛✳s✳✱ ✐✳❡✳ t❤❡ ❢❛✐❧✉r❡ ♣r♦❜❛❜✐❧✐t② P (Fn) t❡♥❞s t♦ ✶ ❛s n t❡♥❞s t♦ ✐♥✜♥✐t②✱ ✷✳ ■❢ ✱ ✱ t❤❡♥ ✱ ✇❤❡r❡ ✐s t❤❡ P♦✐ss♦♥ ❞✐str✐❜✉t✐♦♥ ✈❡❝t♦r ❛♥❞ ✸✳ ■❢ t❤❡♥ ✐s ♥❡❣❧✐❣✐❜❧❡ ✭✐✳❡✳ ❛s ✮ ❛♥❞✱ ♠♦r❡♦✈❡r✱ ❢♦r ✳ ✹✳ ■❢✱ ✐♥ ❝❛s❡ ✸✱ ✐s ❝♦♥st❛♥t✱ t❤❡♥ ✐s ❛s②♠♣t♦t✐❝ t♦ ✳

❙❡❛r❝❤✐♥❣ ❢♦r ❛ ●r❛♣❤ ■s♦♠♦r♣❤✐s♠ ❊✲t❡st

❋❛✐❧✉r❡ ♣r♦❜❛❜✐❧✐t②

◮ ❚❤❡♦r❡♠ ✭●✳❆✳◆♦s❦♦✈✲❆✳◆✳❘②❜❛❧♦✈✱✷✵✶✹✮

▲❡t Fn ❜❡ t❤❡ ❢❛✐❧✉r❡ s❡t ♦❢ t❤❡ ❊✲t❡st ♦♥ t❤❡ s❡t ♦❢ ✐♥♣✉ts G (n, p) × G (n, p) , n ∈ N✱ 0 < p ≤ 1

2✳

✶✳ ■❢ limn→∞ np = 0 , t❤❡♥ F = ∪Fn ❤❛♣♣❡♥s ❛✳❛✳s✳✱ ✐✳❡✳ t❤❡ ❢❛✐❧✉r❡ ♣r♦❜❛❜✐❧✐t② P (Fn) t❡♥❞s t♦ ✶ ❛s n t❡♥❞s t♦ ✐♥✜♥✐t②✱ ✷✳ ■❢ limn→∞ np = λ✱ λ > 0 ✱ t❤❡♥ P (Fn) = Π (λ)2

2✱ ✇❤❡r❡ Π (λ)

✐s t❤❡ P♦✐ss♦♥ ❞✐str✐❜✉t✐♦♥ ✈❡❝t♦r ❛♥❞ 0 < Π (λ)2

2 < min

• eλ2−2λ, 1
• ,

✸✳ ■❢ t❤❡♥ ✐s ♥❡❣❧✐❣✐❜❧❡ ✭✐✳❡✳ ❛s ✮ ❛♥❞✱ ♠♦r❡♦✈❡r✱ ❢♦r ✳ ✹✳ ■❢✱ ✐♥ ❝❛s❡ ✸✱ ✐s ❝♦♥st❛♥t✱ t❤❡♥ ✐s ❛s②♠♣t♦t✐❝ t♦ ✳

❙❡❛r❝❤✐♥❣ ❢♦r ❛ ●r❛♣❤ ■s♦♠♦r♣❤✐s♠ ❊✲t❡st

❋❛✐❧✉r❡ ♣r♦❜❛❜✐❧✐t②

◮ ❚❤❡♦r❡♠ ✭●✳❆✳◆♦s❦♦✈✲❆✳◆✳❘②❜❛❧♦✈✱✷✵✶✹✮

▲❡t Fn ❜❡ t❤❡ ❢❛✐❧✉r❡ s❡t ♦❢ t❤❡ ❊✲t❡st ♦♥ t❤❡ s❡t ♦❢ ✐♥♣✉ts G (n, p) × G (n, p) , n ∈ N✱ 0 < p ≤ 1

2✳

✶✳ ■❢ limn→∞ np = 0 , t❤❡♥ F = ∪Fn ❤❛♣♣❡♥s ❛✳❛✳s✳✱ ✐✳❡✳ t❤❡ ❢❛✐❧✉r❡ ♣r♦❜❛❜✐❧✐t② P (Fn) t❡♥❞s t♦ ✶ ❛s n t❡♥❞s t♦ ✐♥✜♥✐t②✱ ✷✳ ■❢ limn→∞ np = λ✱ λ > 0 ✱ t❤❡♥ P (Fn) = Π (λ)2

2✱ ✇❤❡r❡ Π (λ)

✐s t❤❡ P♦✐ss♦♥ ❞✐str✐❜✉t✐♦♥ ✈❡❝t♦r ❛♥❞ 0 < Π (λ)2

2 < min

• eλ2−2λ, 1
• ,

✸✳ ■❢ limn→∞ np = ∞, t❤❡♥ F ✐s ♥❡❣❧✐❣✐❜❧❡ ✭✐✳❡✳ P (Fn) → 0 ❛s n → ∞✮ ❛♥❞✱ ♠♦r❡♦✈❡r✱ P (Fn) ≤

10 √Npq ❢♦r N =

n

2

• ≥ 10✳

✹✳ ■❢✱ ✐♥ ❝❛s❡ ✸✱ ✐s ❝♦♥st❛♥t✱ t❤❡♥ ✐s ❛s②♠♣t♦t✐❝ t♦ ✳

❙❡❛r❝❤✐♥❣ ❢♦r ❛ ●r❛♣❤ ■s♦♠♦r♣❤✐s♠ ❊✲t❡st

❋❛✐❧✉r❡ ♣r♦❜❛❜✐❧✐t②

◮ ❚❤❡♦r❡♠ ✭●✳❆✳◆♦s❦♦✈✲❆✳◆✳❘②❜❛❧♦✈✱✷✵✶✹✮

▲❡t Fn ❜❡ t❤❡ ❢❛✐❧✉r❡ s❡t ♦❢ t❤❡ ❊✲t❡st ♦♥ t❤❡ s❡t ♦❢ ✐♥♣✉ts G (n, p) × G (n, p) , n ∈ N✱ 0 < p ≤ 1

2✳

✶✳ ■❢ limn→∞ np = 0 , t❤❡♥ F = ∪Fn ❤❛♣♣❡♥s ❛✳❛✳s✳✱ ✐✳❡✳ t❤❡ ❢❛✐❧✉r❡ ♣r♦❜❛❜✐❧✐t② P (Fn) t❡♥❞s t♦ ✶ ❛s n t❡♥❞s t♦ ✐♥✜♥✐t②✱ ✷✳ ■❢ limn→∞ np = λ✱ λ > 0 ✱ t❤❡♥ P (Fn) = Π (λ)2

2✱ ✇❤❡r❡ Π (λ)

✐s t❤❡ P♦✐ss♦♥ ❞✐str✐❜✉t✐♦♥ ✈❡❝t♦r ❛♥❞ 0 < Π (λ)2

2 < min

• eλ2−2λ, 1
• ,

✸✳ ■❢ limn→∞ np = ∞, t❤❡♥ F ✐s ♥❡❣❧✐❣✐❜❧❡ ✭✐✳❡✳ P (Fn) → 0 ❛s n → ∞✮ ❛♥❞✱ ♠♦r❡♦✈❡r✱ P (Fn) ≤

10 √Npq ❢♦r N =

n

2

• ≥ 10✳

✹✳ ■❢✱ ✐♥ ❝❛s❡ ✸✱ p ✐s ❝♦♥st❛♥t✱ t❤❡♥ P (Fn) ✐s ❛s②♠♣t♦t✐❝ t♦

1 n√2πpq✳

❙❡❛r❝❤✐♥❣ ❢♦r ❛ ●r❛♣❤ ■s♦♠♦r♣❤✐s♠ ❊✲t❡st

❋❛✐❧✉r❡ ♣r♦❜❛❜✐❧✐t②

◮ ❚❤❡♦r❡♠ ✭●✳❆✳◆♦s❦♦✈✲❆✳◆✳❘②❜❛❧♦✈✱✷✵✶✹✮

▲❡t Fn ❜❡ t❤❡ ❢❛✐❧✉r❡ s❡t ♦❢ t❤❡ ❊✲t❡st ♦♥ t❤❡ s❡t ♦❢ ✐♥♣✉ts G (n, p) × G (n, p) , n ∈ N✱ 0 < p ≤ 1

2✳

✶✳ ■❢ limn→∞ np = 0 , t❤❡♥ F = ∪Fn ❤❛♣♣❡♥s ❛✳❛✳s✳✱ ✐✳❡✳ t❤❡ ❢❛✐❧✉r❡ ♣r♦❜❛❜✐❧✐t② P (Fn) t❡♥❞s t♦ ✶ ❛s n t❡♥❞s t♦ ✐♥✜♥✐t②✱ ✷✳ ■❢ limn→∞ np = λ✱ λ > 0 ✱ t❤❡♥ P (Fn) = Π (λ)2

2✱ ✇❤❡r❡ Π (λ)

✐s t❤❡ P♦✐ss♦♥ ❞✐str✐❜✉t✐♦♥ ✈❡❝t♦r ❛♥❞ 0 < Π (λ)2

2 < min

• eλ2−2λ, 1
• ,

✸✳ ■❢ limn→∞ np = ∞, t❤❡♥ F ✐s ♥❡❣❧✐❣✐❜❧❡ ✭✐✳❡✳ P (Fn) → 0 ❛s n → ∞✮ ❛♥❞✱ ♠♦r❡♦✈❡r✱ P (Fn) ≤

10 √Npq ❢♦r N =

n

2

• ≥ 10✳

✹✳ ■❢✱ ✐♥ ❝❛s❡ ✸✱ p ✐s ❝♦♥st❛♥t✱ t❤❡♥ P (Fn) ✐s ❛s②♠♣t♦t✐❝ t♦

1 n√2πpq✳

❙❡❛r❝❤✐♥❣ ❢♦r ❛ ●r❛♣❤ ■s♦♠♦r♣❤✐s♠ ❊✲t❡st

Pr♦♦❢✿ ❈❛s❡ np(n) → λ > 0

◮ ❇② t❤❡ P♦✐ss♦♥ t❤❡♦r❡♠ ❢♦r ❡✈❡r② ♥❛t✉r❛❧ k

b (k; n, p) → πk, n → ∞, ✇❤❡r❡ ❚❤❛t ✐s✱ ❝♦♦r❞✐♥❛t❡✇✐s❡✱ ✇❤❡r❡ ✐s t❤❡ P♦✐ss♦♥ ❞✐str✐❜✉t✐♦♥ ✈❡❝t♦r✳ ❯s✐♥❣ t❤❡ r❡s✉❧t ♦❢ ❨✉✳❱✳Pr♦❤♦r♦✈ ✭✶✾✺✸✮ ✇❡ ❞❡❞✉❝❡

❙❡❛r❝❤✐♥❣ ❢♦r ❛ ●r❛♣❤ ■s♦♠♦r♣❤✐s♠ ❊✲t❡st

Pr♦♦❢✿ ❈❛s❡ np(n) → λ > 0

◮ ❇② t❤❡ P♦✐ss♦♥ t❤❡♦r❡♠ ❢♦r ❡✈❡r② ♥❛t✉r❛❧ k

b (k; n, p) → πk, n → ∞,

◮ ✇❤❡r❡

πk = e−λ λk k! , k = 0, 1, 2, . . . . ❚❤❛t ✐s✱ ❝♦♦r❞✐♥❛t❡✇✐s❡✱ ✇❤❡r❡ ✐s t❤❡ P♦✐ss♦♥ ❞✐str✐❜✉t✐♦♥ ✈❡❝t♦r✳ ❯s✐♥❣ t❤❡ r❡s✉❧t ♦❢ ❨✉✳❱✳Pr♦❤♦r♦✈ ✭✶✾✺✸✮ ✇❡ ❞❡❞✉❝❡

❙❡❛r❝❤✐♥❣ ❢♦r ❛ ●r❛♣❤ ■s♦♠♦r♣❤✐s♠ ❊✲t❡st

Pr♦♦❢✿ ❈❛s❡ np(n) → λ > 0

◮ ❇② t❤❡ P♦✐ss♦♥ t❤❡♦r❡♠ ❢♦r ❡✈❡r② ♥❛t✉r❛❧ k

b (k; n, p) → πk, n → ∞,

◮ ✇❤❡r❡

πk = e−λ λk k! , k = 0, 1, 2, . . . .

◮ ❚❤❛t ✐s✱ B (n, p) → Π (λ) ❝♦♦r❞✐♥❛t❡✇✐s❡✱ ✇❤❡r❡

Π (λ) = (π0 (λ) , ..., πk (λ) , ...) , ✐s t❤❡ P♦✐ss♦♥ ❞✐str✐❜✉t✐♦♥ ✈❡❝t♦r✳ ❯s✐♥❣ t❤❡ r❡s✉❧t ♦❢ ❨✉✳❱✳Pr♦❤♦r♦✈ ✭✶✾✺✸✮ ✇❡ ❞❡❞✉❝❡

❙❡❛r❝❤✐♥❣ ❢♦r ❛ ●r❛♣❤ ■s♦♠♦r♣❤✐s♠ ❊✲t❡st

Pr♦♦❢✿ ❈❛s❡ np(n) → λ > 0

◮ ❇② t❤❡ P♦✐ss♦♥ t❤❡♦r❡♠ ❢♦r ❡✈❡r② ♥❛t✉r❛❧ k

b (k; n, p) → πk, n → ∞,

◮ ✇❤❡r❡

πk = e−λ λk k! , k = 0, 1, 2, . . . .

◮ ❚❤❛t ✐s✱ B (n, p) → Π (λ) ❝♦♦r❞✐♥❛t❡✇✐s❡✱ ✇❤❡r❡

Π (λ) = (π0 (λ) , ..., πk (λ) , ...) , ✐s t❤❡ P♦✐ss♦♥ ❞✐str✐❜✉t✐♦♥ ✈❡❝t♦r✳

◮ ❯s✐♥❣ t❤❡ r❡s✉❧t ♦❢ ❨✉✳❱✳Pr♦❤♦r♦✈ ✭✶✾✺✸✮ ✇❡ ❞❡❞✉❝❡

lim

n→∞ |B (n, p)|2 = |Π (λ)|2 .

❙❡❛r❝❤✐♥❣ ❢♦r ❛ ●r❛♣❤ ■s♦♠♦r♣❤✐s♠ ❊✲t❡st

Pr♦♦❢✿ ❈❛s❡ np(n) → λ > 0

◮ ❇② t❤❡ P♦✐ss♦♥ t❤❡♦r❡♠ ❢♦r ❡✈❡r② ♥❛t✉r❛❧ k

b (k; n, p) → πk, n → ∞,

◮ ✇❤❡r❡

πk = e−λ λk k! , k = 0, 1, 2, . . . .

◮ ❚❤❛t ✐s✱ B (n, p) → Π (λ) ❝♦♦r❞✐♥❛t❡✇✐s❡✱ ✇❤❡r❡

Π (λ) = (π0 (λ) , ..., πk (λ) , ...) , ✐s t❤❡ P♦✐ss♦♥ ❞✐str✐❜✉t✐♦♥ ✈❡❝t♦r✳

◮ ❯s✐♥❣ t❤❡ r❡s✉❧t ♦❢ ❨✉✳❱✳Pr♦❤♦r♦✈ ✭✶✾✺✸✮ ✇❡ ❞❡❞✉❝❡

lim

n→∞ |B (n, p)|2 = |Π (λ)|2 .

❙❡❛r❝❤✐♥❣ ❢♦r ❛ ●r❛♣❤ ■s♦♠♦r♣❤✐s♠ ❊✲t❡st

❈❛s❡ p = const

✇❤❡r❡ ✳

❙❡❛r❝❤✐♥❣ ❢♦r ❛ ●r❛♣❤ ■s♦♠♦r♣❤✐s♠ ❊✲t❡st

❈❛s❡ p = const

◮

Pn (Fn) =

N

• i=0

N i 2 p2iq2N−2i = q2NSN p q 2 , ✇❤❡r❡ Sm (t) = m

i=0

m

i

2ti✳

❙❡❛r❝❤✐♥❣ ❢♦r ❛ ●r❛♣❤ ■s♦♠♦r♣❤✐s♠ ❊✲t❡st

❈❛s❡ p = const

◮

Pn (Fn) =

N

• i=0

N i 2 p2iq2N−2i = q2NSN p q 2 , ✇❤❡r❡ Sm (t) = m

i=0

m

i

2ti✳

❙❡❛r❝❤✐♥❣ ❢♦r ❛ ●r❛♣❤ ■s♦♠♦r♣❤✐s♠ ❊✲t❡st

❈❛s❡ p = const

◮ ■♥ ✐ts t✉r♥ t❤❡ q✉❛❞r❛t✐❝ ❜✐♥♦♠✐❛❧ s✉♠ ❝❛♥ ❜❡ ❡①♣r❡ss❡❞ ✐♥

t❡r♠s ♦❢ ▲❡❣❡♥❞r❡ ♣♦❧②♥♦♠✐❛❧✳ Pr❡❝✐s❡❧②✱ s❡tt✐♥❣ t =

• p

q

2 ✇❡ ♦❜t❛✐♥ ✇❤❡r❡

❙❡❛r❝❤✐♥❣ ❢♦r ❛ ●r❛♣❤ ■s♦♠♦r♣❤✐s♠ ❊✲t❡st

❈❛s❡ p = const

◮ ■♥ ✐ts t✉r♥ t❤❡ q✉❛❞r❛t✐❝ ❜✐♥♦♠✐❛❧ s✉♠ ❝❛♥ ❜❡ ❡①♣r❡ss❡❞ ✐♥

t❡r♠s ♦❢ ▲❡❣❡♥❞r❡ ♣♦❧②♥♦♠✐❛❧✳ Pr❡❝✐s❡❧②✱ s❡tt✐♥❣ t =

• p

q

2 ✇❡ ♦❜t❛✐♥

◮

Pn (Fn) = (1 − 2p)N LN   1 +

• p2

q2

• 1 −
• p2

q2

• 

 , ✇❤❡r❡

❙❡❛r❝❤✐♥❣ ❢♦r ❛ ●r❛♣❤ ■s♦♠♦r♣❤✐s♠ ❊✲t❡st

❈❛s❡ p = const

◮ ■♥ ✐ts t✉r♥ t❤❡ q✉❛❞r❛t✐❝ ❜✐♥♦♠✐❛❧ s✉♠ ❝❛♥ ❜❡ ❡①♣r❡ss❡❞ ✐♥

t❡r♠s ♦❢ ▲❡❣❡♥❞r❡ ♣♦❧②♥♦♠✐❛❧✳ Pr❡❝✐s❡❧②✱ s❡tt✐♥❣ t =

• p

q

2 ✇❡ ♦❜t❛✐♥

◮

Pn (Fn) = (1 − 2p)N LN   1 +

• p2

q2

• 1 −
• p2

q2

• 

 , ✇❤❡r❡

◮

Ln (x) = 1 2nn! dn dxn

• x2 − 1

n , (n ≥ 1) .

❙❡❛r❝❤✐♥❣ ❢♦r ❛ ●r❛♣❤ ■s♦♠♦r♣❤✐s♠ ❊✲t❡st

❈❛s❡ p = const

◮ ■♥ ✐ts t✉r♥ t❤❡ q✉❛❞r❛t✐❝ ❜✐♥♦♠✐❛❧ s✉♠ ❝❛♥ ❜❡ ❡①♣r❡ss❡❞ ✐♥

t❡r♠s ♦❢ ▲❡❣❡♥❞r❡ ♣♦❧②♥♦♠✐❛❧✳ Pr❡❝✐s❡❧②✱ s❡tt✐♥❣ t =

• p

q

2 ✇❡ ♦❜t❛✐♥

◮

Pn (Fn) = (1 − 2p)N LN   1 +

• p2

q2

• 1 −
• p2

q2

• 

 , ✇❤❡r❡

◮

Ln (x) = 1 2nn! dn dxn

• x2 − 1

n , (n ≥ 1) .

❙❡❛r❝❤✐♥❣ ❢♦r ❛ ●r❛♣❤ ■s♦♠♦r♣❤✐s♠ ❊✲t❡st

❈❛s❡ p = const

◮ ❚♦ ❞❡t❡r♠✐♥❡ t❤❡ ❛s②♠♣t♦t✐❝ ♦❢ LN

• 1+
• p

q

2 1−

• p

q

2

• ✇❡ ✉s❡ t❤❡

▲❛♣❧❛❝❡✲❍❡✐♥❡ ❢♦r♠✉❧❛ LN (x) ∼ 1 √ 2πN

4

√ x2 − 1

• x +
• x2 − 1

N+ 1

2 ,

(|x| > 1)

❙❡❛r❝❤✐♥❣ ❢♦r ❛ ●r❛♣❤ ■s♦♠♦r♣❤✐s♠ ❊✲t❡st

❈❛s❡ p = const

◮ ❚♦ ❞❡t❡r♠✐♥❡ t❤❡ ❛s②♠♣t♦t✐❝ ♦❢ LN

• 1+
• p

q

2 1−

• p

q

2

• ✇❡ ✉s❡ t❤❡

▲❛♣❧❛❝❡✲❍❡✐♥❡ ❢♦r♠✉❧❛ LN (x) ∼ 1 √ 2πN

4

√ x2 − 1

• x +
• x2 − 1

N+ 1

2 ,

(|x| > 1)

❙❡❛r❝❤✐♥❣ ❢♦r ❛ ●r❛♣❤ ■s♦♠♦r♣❤✐s♠ ◗✉❡st✐♦♥s

◗✉❡st✐♦♥s

◮ ❲❤❛t ✐s t❤❡ ❛s②♠♣t♦t✐❝ ♦❢ t❤❡ ❢❛✐❧✉r❡ ♣r♦❜❛❜✐❧✐t② ♦❢ t❤❡ ❊✲t❡st

✐♥ ❝❛s❡ ♦❢ ♣❧❛♥❛r ❣r❛♣❤s✱ ❣r❛♣❤s ♦❢ ❜♦✉♥❞❡❞ ❞❡❣r❡❡✱ s♦♠❡ ♦t❤❡r ✐♥t❡r❡st✐♥❣ ❝❧❛ss❡s ♦❢ ❣r❛♣❤s❄ ❚❤❡ ♠♦❞❡❧ ❝♦♥s✐sts ♦❢ ❣r❛♣❤s ♦♥ t❤❡ s❡t ✇✐t❤ ❡❞❣❡s✳ ❚❤❡ ❡❞❣❡ t❡st ❢❛✐❧s ❤❡r❡ ❢r♦♠ t❤❡ ✈❡r② ❜❡❣✐♥♥✐♥❣✳ ❚❤❡ r✐❣❤t ♠♦❞✐✜❝❛t✐♦♥ ✇♦✉❧❞ ❜❡ ❛ t❡st✱ ❜❛s❡❞ ♦♥ ❝♦✉♥t✐♥❣ t❤❡ ♥✉♠❜❡r ♦❢ ✷✲✇❛❧❦s✳ ❲❤❛t ✐s ✐ts ❢❛✐❧✉r❡ ♣r♦❜❛❜✐❧✐t②❄ ◆♦t❡✱ t❤❛t t❤❡ ♥✉♠❜❡r ♦❢ ✷✲✇❛❧❦s ❡q✉❛❧s ✇❤❡r❡ ✐s t❤❡ ❞❡❣r❡❡ ♦❢ t❤❡ ✈❡rt❡① ✳ ❲❤❛t ✐s t❤❡ ❛s②♠♣t♦t✐❝ ❢❛✐❧✉r❡ ♣r♦❜❛❜✐❧✐t② ♦❢ t❡sts ❜❛s❡❞ ♦♥ ❝♦✉♥t✐♥❣ ♦❢ ✲✇❛❧❦s ✇✐t❤ ✜①❡❞ ❄ ❚❤❡ s❛♠❡ ✇✐t❤ ❝♦✉♥t✐♥❣ ❛❧❧ ✇❛❧❦s❄

❙❡❛r❝❤✐♥❣ ❢♦r ❛ ●r❛♣❤ ■s♦♠♦r♣❤✐s♠ ◗✉❡st✐♦♥s

◗✉❡st✐♦♥s

◮ ❲❤❛t ✐s t❤❡ ❛s②♠♣t♦t✐❝ ♦❢ t❤❡ ❢❛✐❧✉r❡ ♣r♦❜❛❜✐❧✐t② ♦❢ t❤❡ ❊✲t❡st

✐♥ ❝❛s❡ ♦❢ ♣❧❛♥❛r ❣r❛♣❤s✱ ❣r❛♣❤s ♦❢ ❜♦✉♥❞❡❞ ❞❡❣r❡❡✱ s♦♠❡ ♦t❤❡r ✐♥t❡r❡st✐♥❣ ❝❧❛ss❡s ♦❢ ❣r❛♣❤s❄

◮ ❚❤❡ ♠♦❞❡❧ G (n, m) ❝♦♥s✐sts ♦❢ ❣r❛♣❤s ♦♥ t❤❡ s❡t [n] ✇✐t❤ m

❡❞❣❡s✳ ❚❤❡ ❡❞❣❡ t❡st ❢❛✐❧s ❤❡r❡ ❢r♦♠ t❤❡ ✈❡r② ❜❡❣✐♥♥✐♥❣✳ ❚❤❡ r✐❣❤t ♠♦❞✐✜❝❛t✐♦♥ ✇♦✉❧❞ ❜❡ ❛ t❡st✱ ❜❛s❡❞ ♦♥ ❝♦✉♥t✐♥❣ t❤❡ ♥✉♠❜❡r ♦❢ ✷✲✇❛❧❦s✳ ❲❤❛t ✐s ✐ts ❢❛✐❧✉r❡ ♣r♦❜❛❜✐❧✐t②❄ ◆♦t❡✱ t❤❛t t❤❡ ♥✉♠❜❡r ♦❢ ✷✲✇❛❧❦s ❡q✉❛❧s n

i=1

di

2

• , ✇❤❡r❡ di ✐s t❤❡

❞❡❣r❡❡ ♦❢ t❤❡ ✈❡rt❡① i✳ ❲❤❛t ✐s t❤❡ ❛s②♠♣t♦t✐❝ ❢❛✐❧✉r❡ ♣r♦❜❛❜✐❧✐t② ♦❢ t❡sts ❜❛s❡❞ ♦♥ ❝♦✉♥t✐♥❣ ♦❢ ✲✇❛❧❦s ✇✐t❤ ✜①❡❞ ❄ ❚❤❡ s❛♠❡ ✇✐t❤ ❝♦✉♥t✐♥❣ ❛❧❧ ✇❛❧❦s❄

❙❡❛r❝❤✐♥❣ ❢♦r ❛ ●r❛♣❤ ■s♦♠♦r♣❤✐s♠ ◗✉❡st✐♦♥s

◗✉❡st✐♦♥s

◮ ❲❤❛t ✐s t❤❡ ❛s②♠♣t♦t✐❝ ♦❢ t❤❡ ❢❛✐❧✉r❡ ♣r♦❜❛❜✐❧✐t② ♦❢ t❤❡ ❊✲t❡st

✐♥ ❝❛s❡ ♦❢ ♣❧❛♥❛r ❣r❛♣❤s✱ ❣r❛♣❤s ♦❢ ❜♦✉♥❞❡❞ ❞❡❣r❡❡✱ s♦♠❡ ♦t❤❡r ✐♥t❡r❡st✐♥❣ ❝❧❛ss❡s ♦❢ ❣r❛♣❤s❄

◮ ❚❤❡ ♠♦❞❡❧ G (n, m) ❝♦♥s✐sts ♦❢ ❣r❛♣❤s ♦♥ t❤❡ s❡t [n] ✇✐t❤ m

❡❞❣❡s✳ ❚❤❡ ❡❞❣❡ t❡st ❢❛✐❧s ❤❡r❡ ❢r♦♠ t❤❡ ✈❡r② ❜❡❣✐♥♥✐♥❣✳ ❚❤❡ r✐❣❤t ♠♦❞✐✜❝❛t✐♦♥ ✇♦✉❧❞ ❜❡ ❛ t❡st✱ ❜❛s❡❞ ♦♥ ❝♦✉♥t✐♥❣ t❤❡ ♥✉♠❜❡r ♦❢ ✷✲✇❛❧❦s✳ ❲❤❛t ✐s ✐ts ❢❛✐❧✉r❡ ♣r♦❜❛❜✐❧✐t②❄ ◆♦t❡✱ t❤❛t t❤❡ ♥✉♠❜❡r ♦❢ ✷✲✇❛❧❦s ❡q✉❛❧s n

i=1

di

2

• , ✇❤❡r❡ di ✐s t❤❡

❞❡❣r❡❡ ♦❢ t❤❡ ✈❡rt❡① i✳

◮ ❲❤❛t ✐s t❤❡ ❛s②♠♣t♦t✐❝ ❢❛✐❧✉r❡ ♣r♦❜❛❜✐❧✐t② ♦❢ t❡sts ❜❛s❡❞ ♦♥

❝♦✉♥t✐♥❣ ♦❢ k✲✇❛❧❦s ✇✐t❤ ✜①❡❞ k ≥ 2 ❄ ❚❤❡ s❛♠❡ ✇✐t❤ ❝♦✉♥t✐♥❣ ❛❧❧ ✇❛❧❦s❄

❙❡❛r❝❤✐♥❣ ❢♦r ❛ ●r❛♣❤ ■s♦♠♦r♣❤✐s♠ ◗✉❡st✐♦♥s

◗✉❡st✐♦♥s

◮ ❲❤❛t ✐s t❤❡ ❛s②♠♣t♦t✐❝ ♦❢ t❤❡ ❢❛✐❧✉r❡ ♣r♦❜❛❜✐❧✐t② ♦❢ t❤❡ ❊✲t❡st

✐♥ ❝❛s❡ ♦❢ ♣❧❛♥❛r ❣r❛♣❤s✱ ❣r❛♣❤s ♦❢ ❜♦✉♥❞❡❞ ❞❡❣r❡❡✱ s♦♠❡ ♦t❤❡r ✐♥t❡r❡st✐♥❣ ❝❧❛ss❡s ♦❢ ❣r❛♣❤s❄

◮ ❚❤❡ ♠♦❞❡❧ G (n, m) ❝♦♥s✐sts ♦❢ ❣r❛♣❤s ♦♥ t❤❡ s❡t [n] ✇✐t❤ m

❡❞❣❡s✳ ❚❤❡ ❡❞❣❡ t❡st ❢❛✐❧s ❤❡r❡ ❢r♦♠ t❤❡ ✈❡r② ❜❡❣✐♥♥✐♥❣✳ ❚❤❡ r✐❣❤t ♠♦❞✐✜❝❛t✐♦♥ ✇♦✉❧❞ ❜❡ ❛ t❡st✱ ❜❛s❡❞ ♦♥ ❝♦✉♥t✐♥❣ t❤❡ ♥✉♠❜❡r ♦❢ ✷✲✇❛❧❦s✳ ❲❤❛t ✐s ✐ts ❢❛✐❧✉r❡ ♣r♦❜❛❜✐❧✐t②❄ ◆♦t❡✱ t❤❛t t❤❡ ♥✉♠❜❡r ♦❢ ✷✲✇❛❧❦s ❡q✉❛❧s n

i=1

di

2

• , ✇❤❡r❡ di ✐s t❤❡

❞❡❣r❡❡ ♦❢ t❤❡ ✈❡rt❡① i✳

◮ ❲❤❛t ✐s t❤❡ ❛s②♠♣t♦t✐❝ ❢❛✐❧✉r❡ ♣r♦❜❛❜✐❧✐t② ♦❢ t❡sts ❜❛s❡❞ ♦♥

❝♦✉♥t✐♥❣ ♦❢ k✲✇❛❧❦s ✇✐t❤ ✜①❡❞ k ≥ 2 ❄ ❚❤❡ s❛♠❡ ✇✐t❤ ❝♦✉♥t✐♥❣ ❛❧❧ ✇❛❧❦s❄

❙❡❛r❝❤✐♥❣ ❢♦r ❛ ●r❛♣❤ ■s♦♠♦r♣❤✐s♠ ◗✉❡st✐♦♥s

◗✉❡st✐♦♥s

◮ ■s t❤❡r❡ ❛ ✧tr✐✈✐❛❧ t❡st✧❢♦r t❤❡ ▼❛①✐♠❛❧ ❈❧✐q✉❡ ♣r♦❜❧❡♠ ✭t♦

✇❤✐❝❤ t❤❡ ●■ r❡❞✉❝❡s ♣♦❧②♥♦♠✐❛❧❧②✮✳ ❲❤❛t ✐s t❤❡ ❣❡♥❡r✐❝ ❝❛s❡ ❝♦♠♣❧❡①✐t② ♦❢ ❍✐❞❞❡♥ ❙✉❜❣r♦✉♣ Pr♦❜❧❡♠❄ ■t ✐s ♥♦t tr✉❡ t❤❛t t❤❡ ❣❡♥❡r✐❝ ❝❛s❡ ❝♦♠♣❧❡①✐t② ✐s ✐♥✈❛r✐❛♥t ✉♥❞❡r ♣♦❧②♥♦♠✐❛❧ ❡q✉✐✈❛❧❡♥❝❡✳ ❚❤✉s ✐t ♠❛❦❡s s❡♥s❡ t♦ ❛s❦ ✇❤❡t❤❡r t❤❡ ❢♦❧❧♦✇✐♥❣ ♣r♦❜❧❡♠s ❛❞♠✐t ❣❡♥❡r✐❝❛❧❧② ♣♦❧②♥♦♠✐❛❧ ❛❧❣♦r✐t❤♠s✿ ❙❡❛r❝❤ ■s♦♠♦r♣❤✐s♠✱ ●r❛♣❤ ❆✉t♦♠♦r♣❤✐s♠✱ ❖r❞❡r ♦❢ t❤❡ ❆✉t♦♠♦r♣❤✐s♠ ●r♦✉♣✱ ◆✉♠❜❡r ♦❢ ●r❛♣❤ ■s♦♠♦r♣❤✐s♠s✱ ❉♦✉❜❧❡ ❈♦s❡t ▼❡♠❜❡rs❤✐♣✱ ❈r♦✉♣ ❋❛❝t♦r✐③❛t✐♦♥✱ ●r♦✉♣ ■♥t❡rs❡❝t✐♦♥✱ ❙❡t✇✐s❡ ❙t❛❜✐❧✐③❡r✳ ❚❤❡r❡ ❛r❡ s❡✈❡r❛❧ ✐♥t❡r❡st✐♥❣ ③❡t❛ ❢✉♥❝t✐♦♥s ♦♥ ❣r❛♣❤s ✭❘✐❡♠❛♥♥✱ ❘✉❡❧❧❡✱ ■❤❛r❛✱ ❙❡❧❜❡r❣ ❛♥❞ ♦t❤❡rs✮✳ ❲❤❛t ✐s t❤❡ ❝♦♠♣❧❡①✐t② ♦❢ t❤❡ ✐s♦♠♦r♣❤✐s♠ t❡sts ❜❛s❡❞ ♦♥ t❤❡s❡ ❢✉♥❝t✐♦♥s❄

❙❡❛r❝❤✐♥❣ ❢♦r ❛ ●r❛♣❤ ■s♦♠♦r♣❤✐s♠ ◗✉❡st✐♦♥s

◗✉❡st✐♦♥s

◮ ■s t❤❡r❡ ❛ ✧tr✐✈✐❛❧ t❡st✧❢♦r t❤❡ ▼❛①✐♠❛❧ ❈❧✐q✉❡ ♣r♦❜❧❡♠ ✭t♦

✇❤✐❝❤ t❤❡ ●■ r❡❞✉❝❡s ♣♦❧②♥♦♠✐❛❧❧②✮✳

◮ ❲❤❛t ✐s t❤❡ ❣❡♥❡r✐❝ ❝❛s❡ ❝♦♠♣❧❡①✐t② ♦❢ ❍✐❞❞❡♥ ❙✉❜❣r♦✉♣

Pr♦❜❧❡♠❄ ■t ✐s ♥♦t tr✉❡ t❤❛t t❤❡ ❣❡♥❡r✐❝ ❝❛s❡ ❝♦♠♣❧❡①✐t② ✐s ✐♥✈❛r✐❛♥t ✉♥❞❡r ♣♦❧②♥♦♠✐❛❧ ❡q✉✐✈❛❧❡♥❝❡✳ ❚❤✉s ✐t ♠❛❦❡s s❡♥s❡ t♦ ❛s❦ ✇❤❡t❤❡r t❤❡ ❢♦❧❧♦✇✐♥❣ ♣r♦❜❧❡♠s ❛❞♠✐t ❣❡♥❡r✐❝❛❧❧② ♣♦❧②♥♦♠✐❛❧ ❛❧❣♦r✐t❤♠s✿ ❙❡❛r❝❤ ■s♦♠♦r♣❤✐s♠✱ ●r❛♣❤ ❆✉t♦♠♦r♣❤✐s♠✱ ❖r❞❡r ♦❢ t❤❡ ❆✉t♦♠♦r♣❤✐s♠ ●r♦✉♣✱ ◆✉♠❜❡r ♦❢ ●r❛♣❤ ■s♦♠♦r♣❤✐s♠s✱ ❉♦✉❜❧❡ ❈♦s❡t ▼❡♠❜❡rs❤✐♣✱ ❈r♦✉♣ ❋❛❝t♦r✐③❛t✐♦♥✱ ●r♦✉♣ ■♥t❡rs❡❝t✐♦♥✱ ❙❡t✇✐s❡ ❙t❛❜✐❧✐③❡r✳ ❚❤❡r❡ ❛r❡ s❡✈❡r❛❧ ✐♥t❡r❡st✐♥❣ ③❡t❛ ❢✉♥❝t✐♦♥s ♦♥ ❣r❛♣❤s ✭❘✐❡♠❛♥♥✱ ❘✉❡❧❧❡✱ ■❤❛r❛✱ ❙❡❧❜❡r❣ ❛♥❞ ♦t❤❡rs✮✳ ❲❤❛t ✐s t❤❡ ❝♦♠♣❧❡①✐t② ♦❢ t❤❡ ✐s♦♠♦r♣❤✐s♠ t❡sts ❜❛s❡❞ ♦♥ t❤❡s❡ ❢✉♥❝t✐♦♥s❄

❙❡❛r❝❤✐♥❣ ❢♦r ❛ ●r❛♣❤ ■s♦♠♦r♣❤✐s♠ ◗✉❡st✐♦♥s

◗✉❡st✐♦♥s

◮ ■s t❤❡r❡ ❛ ✧tr✐✈✐❛❧ t❡st✧❢♦r t❤❡ ▼❛①✐♠❛❧ ❈❧✐q✉❡ ♣r♦❜❧❡♠ ✭t♦

✇❤✐❝❤ t❤❡ ●■ r❡❞✉❝❡s ♣♦❧②♥♦♠✐❛❧❧②✮✳

◮ ❲❤❛t ✐s t❤❡ ❣❡♥❡r✐❝ ❝❛s❡ ❝♦♠♣❧❡①✐t② ♦❢ ❍✐❞❞❡♥ ❙✉❜❣r♦✉♣

Pr♦❜❧❡♠❄

◮ ■t ✐s ♥♦t tr✉❡ t❤❛t t❤❡ ❣❡♥❡r✐❝ ❝❛s❡ ❝♦♠♣❧❡①✐t② ✐s ✐♥✈❛r✐❛♥t

✉♥❞❡r ♣♦❧②♥♦♠✐❛❧ ❡q✉✐✈❛❧❡♥❝❡✳ ❚❤✉s ✐t ♠❛❦❡s s❡♥s❡ t♦ ❛s❦ ✇❤❡t❤❡r t❤❡ ❢♦❧❧♦✇✐♥❣ ♣r♦❜❧❡♠s ❛❞♠✐t ❣❡♥❡r✐❝❛❧❧② ♣♦❧②♥♦♠✐❛❧ ❛❧❣♦r✐t❤♠s✿ ❙❡❛r❝❤ ■s♦♠♦r♣❤✐s♠✱ ●r❛♣❤ ❆✉t♦♠♦r♣❤✐s♠✱ ❖r❞❡r ♦❢ t❤❡ ❆✉t♦♠♦r♣❤✐s♠ ●r♦✉♣✱ ◆✉♠❜❡r ♦❢ ●r❛♣❤ ■s♦♠♦r♣❤✐s♠s✱ ❉♦✉❜❧❡ ❈♦s❡t ▼❡♠❜❡rs❤✐♣✱ ❈r♦✉♣ ❋❛❝t♦r✐③❛t✐♦♥✱ ●r♦✉♣ ■♥t❡rs❡❝t✐♦♥✱ ❙❡t✇✐s❡ ❙t❛❜✐❧✐③❡r✳ ❚❤❡r❡ ❛r❡ s❡✈❡r❛❧ ✐♥t❡r❡st✐♥❣ ③❡t❛ ❢✉♥❝t✐♦♥s ♦♥ ❣r❛♣❤s ✭❘✐❡♠❛♥♥✱ ❘✉❡❧❧❡✱ ■❤❛r❛✱ ❙❡❧❜❡r❣ ❛♥❞ ♦t❤❡rs✮✳ ❲❤❛t ✐s t❤❡ ❝♦♠♣❧❡①✐t② ♦❢ t❤❡ ✐s♦♠♦r♣❤✐s♠ t❡sts ❜❛s❡❞ ♦♥ t❤❡s❡ ❢✉♥❝t✐♦♥s❄

❙❡❛r❝❤✐♥❣ ❢♦r ❛ ●r❛♣❤ ■s♦♠♦r♣❤✐s♠ ◗✉❡st✐♦♥s

◗✉❡st✐♦♥s

◮ ■s t❤❡r❡ ❛ ✧tr✐✈✐❛❧ t❡st✧❢♦r t❤❡ ▼❛①✐♠❛❧ ❈❧✐q✉❡ ♣r♦❜❧❡♠ ✭t♦

✇❤✐❝❤ t❤❡ ●■ r❡❞✉❝❡s ♣♦❧②♥♦♠✐❛❧❧②✮✳

◮ ❲❤❛t ✐s t❤❡ ❣❡♥❡r✐❝ ❝❛s❡ ❝♦♠♣❧❡①✐t② ♦❢ ❍✐❞❞❡♥ ❙✉❜❣r♦✉♣

Pr♦❜❧❡♠❄

◮ ■t ✐s ♥♦t tr✉❡ t❤❛t t❤❡ ❣❡♥❡r✐❝ ❝❛s❡ ❝♦♠♣❧❡①✐t② ✐s ✐♥✈❛r✐❛♥t

✉♥❞❡r ♣♦❧②♥♦♠✐❛❧ ❡q✉✐✈❛❧❡♥❝❡✳ ❚❤✉s ✐t ♠❛❦❡s s❡♥s❡ t♦ ❛s❦ ✇❤❡t❤❡r t❤❡ ❢♦❧❧♦✇✐♥❣ ♣r♦❜❧❡♠s ❛❞♠✐t ❣❡♥❡r✐❝❛❧❧② ♣♦❧②♥♦♠✐❛❧ ❛❧❣♦r✐t❤♠s✿ ❙❡❛r❝❤ ■s♦♠♦r♣❤✐s♠✱ ●r❛♣❤ ❆✉t♦♠♦r♣❤✐s♠✱ ❖r❞❡r ♦❢ t❤❡ ❆✉t♦♠♦r♣❤✐s♠ ●r♦✉♣✱ ◆✉♠❜❡r ♦❢ ●r❛♣❤ ■s♦♠♦r♣❤✐s♠s✱ ❉♦✉❜❧❡ ❈♦s❡t ▼❡♠❜❡rs❤✐♣✱ ❈r♦✉♣ ❋❛❝t♦r✐③❛t✐♦♥✱ ●r♦✉♣ ■♥t❡rs❡❝t✐♦♥✱ ❙❡t✇✐s❡ ❙t❛❜✐❧✐③❡r✳

◮ ❚❤❡r❡ ❛r❡ s❡✈❡r❛❧ ✐♥t❡r❡st✐♥❣ ③❡t❛ ❢✉♥❝t✐♦♥s ♦♥ ❣r❛♣❤s

✭❘✐❡♠❛♥♥✱ ❘✉❡❧❧❡✱ ■❤❛r❛✱ ❙❡❧❜❡r❣ ❛♥❞ ♦t❤❡rs✮✳ ❲❤❛t ✐s t❤❡ ❝♦♠♣❧❡①✐t② ♦❢ t❤❡ ✐s♦♠♦r♣❤✐s♠ t❡sts ❜❛s❡❞ ♦♥ t❤❡s❡ ❢✉♥❝t✐♦♥s❄

❙❡❛r❝❤✐♥❣ ❢♦r ❛ ●r❛♣❤ ■s♦♠♦r♣❤✐s♠ ◗✉❡st✐♦♥s

◗✉❡st✐♦♥s

◮ ■s t❤❡r❡ ❛ ✧tr✐✈✐❛❧ t❡st✧❢♦r t❤❡ ▼❛①✐♠❛❧ ❈❧✐q✉❡ ♣r♦❜❧❡♠ ✭t♦

✇❤✐❝❤ t❤❡ ●■ r❡❞✉❝❡s ♣♦❧②♥♦♠✐❛❧❧②✮✳

◮ ❲❤❛t ✐s t❤❡ ❣❡♥❡r✐❝ ❝❛s❡ ❝♦♠♣❧❡①✐t② ♦❢ ❍✐❞❞❡♥ ❙✉❜❣r♦✉♣

Pr♦❜❧❡♠❄

◮ ■t ✐s ♥♦t tr✉❡ t❤❛t t❤❡ ❣❡♥❡r✐❝ ❝❛s❡ ❝♦♠♣❧❡①✐t② ✐s ✐♥✈❛r✐❛♥t

✉♥❞❡r ♣♦❧②♥♦♠✐❛❧ ❡q✉✐✈❛❧❡♥❝❡✳ ❚❤✉s ✐t ♠❛❦❡s s❡♥s❡ t♦ ❛s❦ ✇❤❡t❤❡r t❤❡ ❢♦❧❧♦✇✐♥❣ ♣r♦❜❧❡♠s ❛❞♠✐t ❣❡♥❡r✐❝❛❧❧② ♣♦❧②♥♦♠✐❛❧ ❛❧❣♦r✐t❤♠s✿ ❙❡❛r❝❤ ■s♦♠♦r♣❤✐s♠✱ ●r❛♣❤ ❆✉t♦♠♦r♣❤✐s♠✱ ❖r❞❡r ♦❢ t❤❡ ❆✉t♦♠♦r♣❤✐s♠ ●r♦✉♣✱ ◆✉♠❜❡r ♦❢ ●r❛♣❤ ■s♦♠♦r♣❤✐s♠s✱ ❉♦✉❜❧❡ ❈♦s❡t ▼❡♠❜❡rs❤✐♣✱ ❈r♦✉♣ ❋❛❝t♦r✐③❛t✐♦♥✱ ●r♦✉♣ ■♥t❡rs❡❝t✐♦♥✱ ❙❡t✇✐s❡ ❙t❛❜✐❧✐③❡r✳

◮ ❚❤❡r❡ ❛r❡ s❡✈❡r❛❧ ✐♥t❡r❡st✐♥❣ ③❡t❛ ❢✉♥❝t✐♦♥s ♦♥ ❣r❛♣❤s

✭❘✐❡♠❛♥♥✱ ❘✉❡❧❧❡✱ ■❤❛r❛✱ ❙❡❧❜❡r❣ ❛♥❞ ♦t❤❡rs✮✳ ❲❤❛t ✐s t❤❡ ❝♦♠♣❧❡①✐t② ♦❢ t❤❡ ✐s♦♠♦r♣❤✐s♠ t❡sts ❜❛s❡❞ ♦♥ t❤❡s❡ ❢✉♥❝t✐♦♥s❄

❙❡❛r❝❤✐♥❣ ❢♦r ❛ ●r❛♣❤ ■s♦♠♦r♣❤✐s♠ ❈♦♥❝❧✉s✐♦♥

❈♦♥❝❧✉s✐♦♥

◮ ➴ñòü òàáàê✱ äà íå÷åì íþõàòü✦

❘❡s♣❡❝t ●■✦ ■❧❧ ●■✦ ■♥❢❡❝t ●■✦ ❍❛♣♣② ✻✵t❤ ❇✐rt❤❞❛②✱ ❆❧❡①❡✐✦

❙❡❛r❝❤✐♥❣ ❢♦r ❛ ●r❛♣❤ ■s♦♠♦r♣❤✐s♠ ❈♦♥❝❧✉s✐♦♥

❈♦♥❝❧✉s✐♦♥

◮ ➴ñòü òàáàê✱ äà íå÷åì íþõàòü✦ ◮ ❘❡s♣❡❝t ●■✦ ◮ ■❧❧ ●■✦

■♥❢❡❝t ●■✦ ❍❛♣♣② ✻✵t❤ ❇✐rt❤❞❛②✱ ❆❧❡①❡✐✦

❙❡❛r❝❤✐♥❣ ❢♦r ❛ ●r❛♣❤ ■s♦♠♦r♣❤✐s♠ ❈♦♥❝❧✉s✐♦♥

❈♦♥❝❧✉s✐♦♥

◮ ➴ñòü òàáàê✱ äà íå÷åì íþõàòü✦ ◮ ❘❡s♣❡❝t ●■✦ ◮ ■❧❧ ●■✦

■♥❢❡❝t ●■✦ ❍❛♣♣② ✻✵t❤ ❇✐rt❤❞❛②✱ ❆❧❡①❡✐✦

❙❡❛r❝❤✐♥❣ ❢♦r ❛ ●r❛♣❤ ■s♦♠♦r♣❤✐s♠ ❈♦♥❝❧✉s✐♦♥

❈♦♥❝❧✉s✐♦♥

◮ ➴ñòü òàáàê✱ äà íå÷åì íþõàòü✦ ◮ ❘❡s♣❡❝t ●■✦ ◮ ■❧❧ ●■✦ ◮ ■♥❢❡❝t ●■✦

❍❛♣♣② ✻✵t❤ ❇✐rt❤❞❛②✱ ❆❧❡①❡✐✦

❙❡❛r❝❤✐♥❣ ❢♦r ❛ ●r❛♣❤ ■s♦♠♦r♣❤✐s♠ ❈♦♥❝❧✉s✐♦♥

❈♦♥❝❧✉s✐♦♥

◮ ➴ñòü òàáàê✱ äà íå÷åì íþõàòü✦ ◮ ❘❡s♣❡❝t ●■✦ ◮ ■❧❧ ●■✦ ◮ ■♥❢❡❝t ●■✦ ◮ ❍❛♣♣② ✻✵t❤ ❇✐rt❤❞❛②✱ ❆❧❡①❡✐✦

❙❡❛r❝❤✐♥❣ ❢♦r ❛ ●r❛♣❤ ■s♦♠♦r♣❤✐s♠ ❈♦♥❝❧✉s✐♦♥

❈♦♥❝❧✉s✐♦♥

◮ ➴ñòü òàáàê✱ äà íå÷åì íþõàòü✦ ◮ ❘❡s♣❡❝t ●■✦ ◮ ■❧❧ ●■✦ ◮ ■♥❢❡❝t ●■✦ ◮ ❍❛♣♣② ✻✵t❤ ❇✐rt❤❞❛②✱ ❆❧❡①❡✐✦