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rttrst rs r r rst Prrs t r t
❆❞❡♠✐r ❍✉❥❞✉r♦✈✐➣ ✭❯♥✐✈❡rs✐t② ♦❢ Pr✐♠♦rs❦❛✱ ❙❧♦✈❡♥✐❛✮ ❏♦✐♥t ✇♦r❦ ✇✐t❤ ❊❞✇❛r❞ ❉♦❜s♦♥✱ ▼❛rt✐♥ ▼✐❧❛♥✐↔ ❛♥❞
✶✻✳✵✻✳✷✵✶✺✳
❆❞❡♠✐r ❍✉❥❞✉r♦✈✐➣ ❱❡rt❡①✲tr❛♥s✐t✐✈❡ ❈■❙ ❣r❛♣❤s
❈■❙ ❣r❛♣❤s ❈❤❛r❛❝t❡r✐③❛t✐♦♥ ♦❢ ✈❡rt❡①✲tr❛♥s✐t✐✈❡ ❈■❙ ❣r❛♣❤s ❈❧❛ss✐✜❝❛t✐♦♥ ♦❢ ✈❡rt❡①✲tr❛♥s✐t✐✈❡ ❈■❙ ❣r❛♣❤s ✇✐t❤ ♠❛①✐♠❛❧ ❝❧✐q✉❡s ♦❢ s✐③❡ ✷ ♦r ✸ ❈❧❛ss✐✜❝❛t✐♦♥ ♦❢ ✈❡rt❡①✲tr❛♥s✐t✐✈❡ ❈■❙ ❣r❛♣❤s ♦❢ ✈❛❧❡♥❝② ❛t ♠♦st ✼
❆❞❡♠✐r ❍✉❥❞✉r♦✈✐➣ ❱❡rt❡①✲tr❛♥s✐t✐✈❡ ❈■❙ ❣r❛♣❤s
▲❡t Γ ❞❡♥♦t❡ ❛ ✜♥✐t❡ s✐♠♣❧❡ ✉♥❞✐r❡❝t❡❞ ❣r❛♣❤✳ ❆ ❝❧✐q✉❡ ✭r❡s♣❡❝t✐✈❡❧②✱ ❛ st❛❜❧❡ s❡t✮ ♦❢ Γ ✐s ❛ s❡t ♦❢ ♣❛✐r✇✐s❡ ❛❞❥❛❝❡♥t ✭r❡s♣❡❝t✐✈❡❧②✱ ♥♦♥✲❛❞❥❛❝❡♥t✮ ✈❡rt✐❝❡s✳ ❚❤❡ ✐♥❝❧✉s✐♦♥ ♠❛①✐♠❛❧ ❝❧✐q✉❡s ❛♥❞ st❛❜❧❡ s❡ts ♦❢ ❛r❡ ❝❛❧❧❡❞ ♠❛①✐♠❛❧ ❝❧✐q✉❡s ❛♥❞ ♠❛①✐♠❛❧ st❛❜❧❡ s❡ts r❡s♣❡❝t✐✈❡❧②✳ ❚❤❡ ♠❛①✐♠❛❧ ❝❛r❞✐♥❛❧✐t② ♦❢ ❛ ❝❧✐q✉❡ ✭r❡s♣❡❝t✐✈❡❧② ❛ st❛❜❧❡ s❡t✮ ♦❢ ✐s ❝❛❧❧❡❞ t❤❡ ❝❧✐q✉❡ ✭r❡s♣❡❝t✐✈❡❧② st❛❜✐❧✐t②✮ ♥✉♠❜❡r ❛♥❞ ❞❡♥♦t❡❞ ✭r❡s♣❡❝t✐✈❡❧② ✮✳ ❉❡✜♥✐t✐♦♥ ❆ ❈■❙ ❣r❛♣❤ ✐s ❛ ❣r❛♣❤ ✐♥ ✇❤✐❝❤ ❡✈❡r② ♠❛①✐♠❛❧ st❛❜❧❡ s❡t ❛♥❞ ❡✈❡r② ♠❛①✐♠❛❧ ❝❧✐q✉❡ ✐♥t❡rs❡❝t ✭❈■❙ st❛♥❞s ❢♦r ✏❈❧✐q✉❡s ■♥t❡rs❡❝t ❙t❛❜❧❡ s❡ts✑✮✳
❆❞❡♠✐r ❍✉❥❞✉r♦✈✐➣ ❱❡rt❡①✲tr❛♥s✐t✐✈❡ ❈■❙ ❣r❛♣❤s
▲❡t Γ ❞❡♥♦t❡ ❛ ✜♥✐t❡ s✐♠♣❧❡ ✉♥❞✐r❡❝t❡❞ ❣r❛♣❤✳ ❆ ❝❧✐q✉❡ ✭r❡s♣❡❝t✐✈❡❧②✱ ❛ st❛❜❧❡ s❡t✮ ♦❢ Γ ✐s ❛ s❡t ♦❢ ♣❛✐r✇✐s❡ ❛❞❥❛❝❡♥t ✭r❡s♣❡❝t✐✈❡❧②✱ ♥♦♥✲❛❞❥❛❝❡♥t✮ ✈❡rt✐❝❡s✳ ❚❤❡ ✐♥❝❧✉s✐♦♥ ♠❛①✐♠❛❧ ❝❧✐q✉❡s ❛♥❞ st❛❜❧❡ s❡ts ♦❢ Γ ❛r❡ ❝❛❧❧❡❞ ♠❛①✐♠❛❧ ❝❧✐q✉❡s ❛♥❞ ♠❛①✐♠❛❧ st❛❜❧❡ s❡ts r❡s♣❡❝t✐✈❡❧②✳ ❚❤❡ ♠❛①✐♠❛❧ ❝❛r❞✐♥❛❧✐t② ♦❢ ❛ ❝❧✐q✉❡ ✭r❡s♣❡❝t✐✈❡❧② ❛ st❛❜❧❡ s❡t✮ ♦❢ Γ ✐s ❝❛❧❧❡❞ t❤❡ ❝❧✐q✉❡ ✭r❡s♣❡❝t✐✈❡❧② st❛❜✐❧✐t②✮ ♥✉♠❜❡r ❛♥❞ ❞❡♥♦t❡❞ ω(Γ) ✭r❡s♣❡❝t✐✈❡❧② α(Γ)✮✳ ❉❡✜♥✐t✐♦♥ ❆ ❈■❙ ❣r❛♣❤ ✐s ❛ ❣r❛♣❤ ✐♥ ✇❤✐❝❤ ❡✈❡r② ♠❛①✐♠❛❧ st❛❜❧❡ s❡t ❛♥❞ ❡✈❡r② ♠❛①✐♠❛❧ ❝❧✐q✉❡ ✐♥t❡rs❡❝t ✭❈■❙ st❛♥❞s ❢♦r ✏❈❧✐q✉❡s ■♥t❡rs❡❝t ❙t❛❜❧❡ s❡ts✑✮✳
❆❞❡♠✐r ❍✉❥❞✉r♦✈✐➣ ❱❡rt❡①✲tr❛♥s✐t✐✈❡ ❈■❙ ❣r❛♣❤s
▲❡t Γ ❞❡♥♦t❡ ❛ ✜♥✐t❡ s✐♠♣❧❡ ✉♥❞✐r❡❝t❡❞ ❣r❛♣❤✳ ❆ ❝❧✐q✉❡ ✭r❡s♣❡❝t✐✈❡❧②✱ ❛ st❛❜❧❡ s❡t✮ ♦❢ Γ ✐s ❛ s❡t ♦❢ ♣❛✐r✇✐s❡ ❛❞❥❛❝❡♥t ✭r❡s♣❡❝t✐✈❡❧②✱ ♥♦♥✲❛❞❥❛❝❡♥t✮ ✈❡rt✐❝❡s✳ ❚❤❡ ✐♥❝❧✉s✐♦♥ ♠❛①✐♠❛❧ ❝❧✐q✉❡s ❛♥❞ st❛❜❧❡ s❡ts ♦❢ Γ ❛r❡ ❝❛❧❧❡❞ ♠❛①✐♠❛❧ ❝❧✐q✉❡s ❛♥❞ ♠❛①✐♠❛❧ st❛❜❧❡ s❡ts r❡s♣❡❝t✐✈❡❧②✳ ❚❤❡ ♠❛①✐♠❛❧ ❝❛r❞✐♥❛❧✐t② ♦❢ ❛ ❝❧✐q✉❡ ✭r❡s♣❡❝t✐✈❡❧② ❛ st❛❜❧❡ s❡t✮ ♦❢ Γ ✐s ❝❛❧❧❡❞ t❤❡ ❝❧✐q✉❡ ✭r❡s♣❡❝t✐✈❡❧② st❛❜✐❧✐t②✮ ♥✉♠❜❡r ❛♥❞ ❞❡♥♦t❡❞ ω(Γ) ✭r❡s♣❡❝t✐✈❡❧② α(Γ)✮✳ ❉❡✜♥✐t✐♦♥ ❆ ❈■❙ ❣r❛♣❤ ✐s ❛ ❣r❛♣❤ ✐♥ ✇❤✐❝❤ ❡✈❡r② ♠❛①✐♠❛❧ st❛❜❧❡ s❡t ❛♥❞ ❡✈❡r② ♠❛①✐♠❛❧ ❝❧✐q✉❡ ✐♥t❡rs❡❝t ✭❈■❙ st❛♥❞s ❢♦r ✏❈❧✐q✉❡s ■♥t❡rs❡❝t ❙t❛❜❧❡ s❡ts✑✮✳
❆❞❡♠✐r ❍✉❥❞✉r♦✈✐➣ ❱❡rt❡①✲tr❛♥s✐t✐✈❡ ❈■❙ ❣r❛♣❤s
❆❞❡♠✐r ❍✉❥❞✉r♦✈✐➣ ❱❡rt❡①✲tr❛♥s✐t✐✈❡ ❈■❙ ❣r❛♣❤s
❆ ❣r❛♣❤ ✐s ❈■❙ ✐❢ ❛♥❞ ♦♥❧② ✐❢ ✐ts ❝♦♠♣❧❡♠❡♥t ✐s ❈■❙✳ ❆ ❞✐s❝♦♥♥❡❝t❡❞ ❣r❛♣❤ ✐s ❈■❙ ✐❢ ❛♥❞ ♦♥❧② ✐❢ ❡❛❝❤ ♦❢ ✐ts ❝♦♥♥❡❝t❡❞ ❝♦♠♣♦♥❡♥t ✐s ❈■❙✳ ❋♦r ❡✈❡r② t✇♦ ❣r❛♣❤s
✶ ❛♥❞ ✷✱ t❤❡ ❧❡①✐❝♦❣r❛♣❤✐❝ ♣r♦❞✉❝t ✶ ✷ ✐s ❈■❙ ✐❢ ❛♥❞ ♦♥❧② ✐❢ ✶ ❛♥❞ ✷ ❛r❡ ❈■❙✳
❚❤❡ ❧❡①✐❝♦❣r❛♣❤✐❝ ♣r♦❞✉❝t ♦❢ ❣r❛♣❤s
✶ ❛♥❞ ✷ ✐s t❤❡ ❣r❛♣❤ ✶ ✷
✇✐t❤ ✈❡rt❡① s❡t
✶ ✷ ✱ ✇❤❡r❡ t✇♦ ✈❡rt✐❝❡s
❛♥❞ ❛r❡ ❛❞❥❛❝❡♥t ✐❢ ❛♥❞ ♦♥❧② ✐❢ ❡✐t❤❡r
✶ ♦r
❛♥❞
✷ ✳
❆❞❡♠✐r ❍✉❥❞✉r♦✈✐➣ ❱❡rt❡①✲tr❛♥s✐t✐✈❡ ❈■❙ ❣r❛♣❤s
❆ ❣r❛♣❤ ✐s ❈■❙ ✐❢ ❛♥❞ ♦♥❧② ✐❢ ✐ts ❝♦♠♣❧❡♠❡♥t ✐s ❈■❙✳ ❆ ❞✐s❝♦♥♥❡❝t❡❞ ❣r❛♣❤ ✐s ❈■❙ ✐❢ ❛♥❞ ♦♥❧② ✐❢ ❡❛❝❤ ♦❢ ✐ts ❝♦♥♥❡❝t❡❞ ❝♦♠♣♦♥❡♥t ✐s ❈■❙✳ ❋♦r ❡✈❡r② t✇♦ ❣r❛♣❤s
✶ ❛♥❞ ✷✱ t❤❡ ❧❡①✐❝♦❣r❛♣❤✐❝ ♣r♦❞✉❝t ✶ ✷ ✐s ❈■❙ ✐❢ ❛♥❞ ♦♥❧② ✐❢ ✶ ❛♥❞ ✷ ❛r❡ ❈■❙✳
❚❤❡ ❧❡①✐❝♦❣r❛♣❤✐❝ ♣r♦❞✉❝t ♦❢ ❣r❛♣❤s
✶ ❛♥❞ ✷ ✐s t❤❡ ❣r❛♣❤ ✶ ✷
✇✐t❤ ✈❡rt❡① s❡t
✶ ✷ ✱ ✇❤❡r❡ t✇♦ ✈❡rt✐❝❡s
❛♥❞ ❛r❡ ❛❞❥❛❝❡♥t ✐❢ ❛♥❞ ♦♥❧② ✐❢ ❡✐t❤❡r
✶ ♦r
❛♥❞
✷ ✳
❆❞❡♠✐r ❍✉❥❞✉r♦✈✐➣ ❱❡rt❡①✲tr❛♥s✐t✐✈❡ ❈■❙ ❣r❛♣❤s
❆ ❣r❛♣❤ ✐s ❈■❙ ✐❢ ❛♥❞ ♦♥❧② ✐❢ ✐ts ❝♦♠♣❧❡♠❡♥t ✐s ❈■❙✳ ❆ ❞✐s❝♦♥♥❡❝t❡❞ ❣r❛♣❤ ✐s ❈■❙ ✐❢ ❛♥❞ ♦♥❧② ✐❢ ❡❛❝❤ ♦❢ ✐ts ❝♦♥♥❡❝t❡❞ ❝♦♠♣♦♥❡♥t ✐s ❈■❙✳ ❋♦r ❡✈❡r② t✇♦ ❣r❛♣❤s Γ✶ ❛♥❞ Γ✷✱ t❤❡ ❧❡①✐❝♦❣r❛♣❤✐❝ ♣r♦❞✉❝t Γ✶[Γ✷] ✐s ❈■❙ ✐❢ ❛♥❞ ♦♥❧② ✐❢ Γ✶ ❛♥❞ Γ✷ ❛r❡ ❈■❙✳ ❚❤❡ ❧❡①✐❝♦❣r❛♣❤✐❝ ♣r♦❞✉❝t ♦❢ ❣r❛♣❤s Γ✶ ❛♥❞ Γ✷ ✐s t❤❡ ❣r❛♣❤ Γ✶[Γ✷] ✇✐t❤ ✈❡rt❡① s❡t V (Γ✶) × V (Γ✷)✱ ✇❤❡r❡ t✇♦ ✈❡rt✐❝❡s (u, x) ❛♥❞ (v, y) ❛r❡ ❛❞❥❛❝❡♥t ✐❢ ❛♥❞ ♦♥❧② ✐❢ ❡✐t❤❡r {u, v} ∈ E(Γ✶) ♦r u = v ❛♥❞ {x, y} ∈ E(Γ✷)✳
❆❞❡♠✐r ❍✉❥❞✉r♦✈✐➣ ❱❡rt❡①✲tr❛♥s✐t✐✈❡ ❈■❙ ❣r❛♣❤s
❆ ❣r❛♣❤ Γ ✐s ✇❡❧❧✲❝♦✈❡r❡❞ ✐❢ ❛❧❧ ✐ts ♠❛①✐♠❛❧ st❛❜❧❡ s❡ts ❛r❡ ♦❢ t❤❡ s❛♠❡ s✐③❡✳ ❆ ❣r❛♣❤ Γ ✐s ❝♦✲✇❡❧❧✲❝♦✈❡r❡❞ ✐❢ ❛❧❧ ✐ts ♠❛①✐♠❛❧ ❝❧✐q✉❡s ❛r❡ ♦❢ t❤❡ s❛♠❡ s✐③❡✳ ✭❊q✉✐✈❛❧❡♥t❧②✿ ✐❢ ✐ts ❝♦♠♣❧❡♠❡♥t ✐s ✇❡❧❧✲❝♦✈❡r❡❞✳✮
❆❞❡♠✐r ❍✉❥❞✉r♦✈✐➣ ❱❡rt❡①✲tr❛♥s✐t✐✈❡ ❈■❙ ❣r❛♣❤s
❆ ❣r❛♣❤ ✐s ❝❛❧❧❡❞ ✈❡rt❡①✲tr❛♥s✐t✐✈❡ ✐❢✱ ❢♦r ❡✈❡r② ♣❛✐r ♦❢ ✈❡rt✐❝❡s✱ t❤❡r❡ ❡①✐sts ❛♥ ❛✉t♦♠♦r♣❤✐s♠ ♦❢ t❤❡ ❣r❛♣❤ ♠❛♣♣✐♥❣ ♦♥❡ t♦ t❤❡ ♦t❤❡r✳ ▲❡t ❜❡ ❛ ✜♥✐t❡ ❣r♦✉♣ ✇✐t❤ ✐❞❡♥t✐t② ❡❧❡♠❡♥t ✶✱ ❛♥❞ ❧❡t ✶ ❜❡ s✉❝❤ t❤❛t
✶
❲❡ ❞❡✜♥❡ t❤❡ ❈❛②❧❡② ❣r❛♣❤ ♦♥ t❤❡ ❣r♦✉♣ ✇✐t❤ r❡s♣❡❝t t♦ t❤❡ ❝♦♥♥❡❝t✐♦♥ s❡t ✱ t♦ ❜❡ t❤❡ ❣r❛♣❤ ✇✐t❤ ✈❡rt❡① s❡t ✱ ❛♥❞ ❡❞❣❡s ♦❢ t❤❡ ❢♦r♠ ❢♦r ❛♥❞ ✳ ❆ ❝✐r❝✉❧❛♥t ✐s ❛ ❈❛②❧❡② ❣r❛♣❤ ♦♥ ❛ ❝②❝❧✐❝ ❣r♦✉♣✳ ❊✈❡r② ❈❛②❧❡② ❣r❛♣❤ ✐s ✈❡rt❡①✲tr❛♥s✐t✐✈❡✳ ❚❤❡ ❝♦♥✈❡rs❡ ✐s ♥♦t tr✉❡ ✭s♠❛❧❧❡st ❡①❛♠♣❧❡ ✐s t❤❡ P❡t❡rs❡♥ ❣r❛♣❤✮✳
❆❞❡♠✐r ❍✉❥❞✉r♦✈✐➣ ❱❡rt❡①✲tr❛♥s✐t✐✈❡ ❈■❙ ❣r❛♣❤s
❆ ❣r❛♣❤ ✐s ❝❛❧❧❡❞ ✈❡rt❡①✲tr❛♥s✐t✐✈❡ ✐❢✱ ❢♦r ❡✈❡r② ♣❛✐r ♦❢ ✈❡rt✐❝❡s✱ t❤❡r❡ ❡①✐sts ❛♥ ❛✉t♦♠♦r♣❤✐s♠ ♦❢ t❤❡ ❣r❛♣❤ ♠❛♣♣✐♥❣ ♦♥❡ t♦ t❤❡ ♦t❤❡r✳ ▲❡t G ❜❡ ❛ ✜♥✐t❡ ❣r♦✉♣ ✇✐t❤ ✐❞❡♥t✐t② ❡❧❡♠❡♥t ✶✱ ❛♥❞ ❧❡t S ⊂ G\{✶} ❜❡ s✉❝❤ t❤❛t S−✶ = S. ❲❡ ❞❡✜♥❡ t❤❡ ❈❛②❧❡② ❣r❛♣❤ Cay(G, S) ♦♥ t❤❡ ❣r♦✉♣ G ✇✐t❤ r❡s♣❡❝t t♦ t❤❡ ❝♦♥♥❡❝t✐♦♥ s❡t S✱ t♦ ❜❡ t❤❡ ❣r❛♣❤ ✇✐t❤ ✈❡rt❡① s❡t G✱ ❛♥❞ ❡❞❣❡s ♦❢ t❤❡ ❢♦r♠ {g, gs} ❢♦r g ∈ G ❛♥❞ s ∈ S✳ ❆ ❝✐r❝✉❧❛♥t ✐s ❛ ❈❛②❧❡② ❣r❛♣❤ ♦♥ ❛ ❝②❝❧✐❝ ❣r♦✉♣✳ ❊✈❡r② ❈❛②❧❡② ❣r❛♣❤ ✐s ✈❡rt❡①✲tr❛♥s✐t✐✈❡✳ ❚❤❡ ❝♦♥✈❡rs❡ ✐s ♥♦t tr✉❡ ✭s♠❛❧❧❡st ❡①❛♠♣❧❡ ✐s t❤❡ P❡t❡rs❡♥ ❣r❛♣❤✮✳
❆❞❡♠✐r ❍✉❥❞✉r♦✈✐➣ ❱❡rt❡①✲tr❛♥s✐t✐✈❡ ❈■❙ ❣r❛♣❤s
❆ ❣r❛♣❤ ✐s ❝❛❧❧❡❞ ✈❡rt❡①✲tr❛♥s✐t✐✈❡ ✐❢✱ ❢♦r ❡✈❡r② ♣❛✐r ♦❢ ✈❡rt✐❝❡s✱ t❤❡r❡ ❡①✐sts ❛♥ ❛✉t♦♠♦r♣❤✐s♠ ♦❢ t❤❡ ❣r❛♣❤ ♠❛♣♣✐♥❣ ♦♥❡ t♦ t❤❡ ♦t❤❡r✳ ▲❡t G ❜❡ ❛ ✜♥✐t❡ ❣r♦✉♣ ✇✐t❤ ✐❞❡♥t✐t② ❡❧❡♠❡♥t ✶✱ ❛♥❞ ❧❡t S ⊂ G\{✶} ❜❡ s✉❝❤ t❤❛t S−✶ = S. ❲❡ ❞❡✜♥❡ t❤❡ ❈❛②❧❡② ❣r❛♣❤ Cay(G, S) ♦♥ t❤❡ ❣r♦✉♣ G ✇✐t❤ r❡s♣❡❝t t♦ t❤❡ ❝♦♥♥❡❝t✐♦♥ s❡t S✱ t♦ ❜❡ t❤❡ ❣r❛♣❤ ✇✐t❤ ✈❡rt❡① s❡t G✱ ❛♥❞ ❡❞❣❡s ♦❢ t❤❡ ❢♦r♠ {g, gs} ❢♦r g ∈ G ❛♥❞ s ∈ S✳ ❆ ❝✐r❝✉❧❛♥t ✐s ❛ ❈❛②❧❡② ❣r❛♣❤ ♦♥ ❛ ❝②❝❧✐❝ ❣r♦✉♣✳ ❊✈❡r② ❈❛②❧❡② ❣r❛♣❤ ✐s ✈❡rt❡①✲tr❛♥s✐t✐✈❡✳ ❚❤❡ ❝♦♥✈❡rs❡ ✐s ♥♦t tr✉❡ ✭s♠❛❧❧❡st ❡①❛♠♣❧❡ ✐s t❤❡ P❡t❡rs❡♥ ❣r❛♣❤✮✳
❆❞❡♠✐r ❍✉❥❞✉r♦✈✐➣ ❱❡rt❡①✲tr❛♥s✐t✐✈❡ ❈■❙ ❣r❛♣❤s
❆ ❣r❛♣❤ ✐s ❝❛❧❧❡❞ ✈❡rt❡①✲tr❛♥s✐t✐✈❡ ✐❢✱ ❢♦r ❡✈❡r② ♣❛✐r ♦❢ ✈❡rt✐❝❡s✱ t❤❡r❡ ❡①✐sts ❛♥ ❛✉t♦♠♦r♣❤✐s♠ ♦❢ t❤❡ ❣r❛♣❤ ♠❛♣♣✐♥❣ ♦♥❡ t♦ t❤❡ ♦t❤❡r✳ ▲❡t G ❜❡ ❛ ✜♥✐t❡ ❣r♦✉♣ ✇✐t❤ ✐❞❡♥t✐t② ❡❧❡♠❡♥t ✶✱ ❛♥❞ ❧❡t S ⊂ G\{✶} ❜❡ s✉❝❤ t❤❛t S−✶ = S. ❲❡ ❞❡✜♥❡ t❤❡ ❈❛②❧❡② ❣r❛♣❤ Cay(G, S) ♦♥ t❤❡ ❣r♦✉♣ G ✇✐t❤ r❡s♣❡❝t t♦ t❤❡ ❝♦♥♥❡❝t✐♦♥ s❡t S✱ t♦ ❜❡ t❤❡ ❣r❛♣❤ ✇✐t❤ ✈❡rt❡① s❡t G✱ ❛♥❞ ❡❞❣❡s ♦❢ t❤❡ ❢♦r♠ {g, gs} ❢♦r g ∈ G ❛♥❞ s ∈ S✳ ❆ ❝✐r❝✉❧❛♥t ✐s ❛ ❈❛②❧❡② ❣r❛♣❤ ♦♥ ❛ ❝②❝❧✐❝ ❣r♦✉♣✳ ❊✈❡r② ❈❛②❧❡② ❣r❛♣❤ ✐s ✈❡rt❡①✲tr❛♥s✐t✐✈❡✳ ❚❤❡ ❝♦♥✈❡rs❡ ✐s ♥♦t tr✉❡ ✭s♠❛❧❧❡st ❡①❛♠♣❧❡ ✐s t❤❡ P❡t❡rs❡♥ ❣r❛♣❤✮✳
❆❞❡♠✐r ❍✉❥❞✉r♦✈✐➣ ❱❡rt❡①✲tr❛♥s✐t✐✈❡ ❈■❙ ❣r❛♣❤s
❚❤❡♦r❡♠ ✭❇♦r♦s✱ ●✉r✈✐❝❤✱ ▼✐❧❛♥✐↔✱ ✷✵✶✹✮ ▲❡t Γ ❜❡ ❛ ❝✐r❝✉❧❛♥t ❣r❛♣❤✳ ❚❤❡♥ Γ ✐s ❛ ❈■❙ ❣r❛♣❤ ✐❢ ❛♥❞ ♦♥❧② ✐❢
✶ ❛❧❧ ♠❛①✐♠❛❧ st❛❜❧❡ s❡ts ❛r❡ ♦❢ s✐③❡ α(Γ)✱ ✷ ❛❧❧ ♠❛①✐♠❛❧ ❝❧✐q✉❡s ❛r❡ ♦❢ s✐③❡ ω(Γ)✱ ✸ α(Γ)ω(Γ) = |V (Γ)|✳
❚❤❡♦r❡♠ ✭❉♦❜s♦♥✱ ❍✱ ▼✐❧❛♥✐↔✱ ❱❡rr❡t✱ ✷✵✶✺✮ ▲❡t ❜❡ ❛ ✈❡rt❡①✲tr❛♥s✐t✐✈❡ ❣r❛♣❤✳ ❚❤❡♥ ✐s ❛ ❈■❙ ❣r❛♣❤ ✐❢ ❛♥❞ ♦♥❧② ✐❢
✶ ❛❧❧ ♠❛①✐♠❛❧ st❛❜❧❡ s❡ts ❛r❡ ♦❢ s✐③❡
✱
✷ ❛❧❧ ♠❛①✐♠❛❧ ❝❧✐q✉❡s ❛r❡ ♦❢ s✐③❡
✱
✸
✳
❆❞❡♠✐r ❍✉❥❞✉r♦✈✐➣ ❱❡rt❡①✲tr❛♥s✐t✐✈❡ ❈■❙ ❣r❛♣❤s
❚❤❡♦r❡♠ ✭❇♦r♦s✱ ●✉r✈✐❝❤✱ ▼✐❧❛♥✐↔✱ ✷✵✶✹✮ ▲❡t Γ ❜❡ ❛ ❝✐r❝✉❧❛♥t ❣r❛♣❤✳ ❚❤❡♥ Γ ✐s ❛ ❈■❙ ❣r❛♣❤ ✐❢ ❛♥❞ ♦♥❧② ✐❢
✶ ❛❧❧ ♠❛①✐♠❛❧ st❛❜❧❡ s❡ts ❛r❡ ♦❢ s✐③❡ α(Γ)✱ ✷ ❛❧❧ ♠❛①✐♠❛❧ ❝❧✐q✉❡s ❛r❡ ♦❢ s✐③❡ ω(Γ)✱ ✸ α(Γ)ω(Γ) = |V (Γ)|✳
❚❤❡♦r❡♠ ✭❉♦❜s♦♥✱ ❍✱ ▼✐❧❛♥✐↔✱ ❱❡rr❡t✱ ✷✵✶✺✮ ▲❡t Γ ❜❡ ❛ ✈❡rt❡①✲tr❛♥s✐t✐✈❡ ❣r❛♣❤✳ ❚❤❡♥ Γ ✐s ❛ ❈■❙ ❣r❛♣❤ ✐❢ ❛♥❞ ♦♥❧② ✐❢
✶ ❛❧❧ ♠❛①✐♠❛❧ st❛❜❧❡ s❡ts ❛r❡ ♦❢ s✐③❡ α(Γ)✱ ✷ ❛❧❧ ♠❛①✐♠❛❧ ❝❧✐q✉❡s ❛r❡ ♦❢ s✐③❡ ω(Γ)✱ ✸ α(Γ)ω(Γ) = |V (Γ)|✳ ❆❞❡♠✐r ❍✉❥❞✉r♦✈✐➣ ❱❡rt❡①✲tr❛♥s✐t✐✈❡ ❈■❙ ❣r❛♣❤s
Pr♦♣♦s✐t✐♦♥ ■❢ n ≥ ✶ t❤❡♥ L(Kn,n) ✐s ❛ ❝♦♥♥❡❝t❡❞ ✈❡rt❡①✲tr❛♥s✐t✐✈❡ ❈■❙ ❣r❛♣❤ ♦❢ ♦r❞❡r n✷ ❛♥❞ ✈❛❧❡♥❝② ✷(n − ✶) ✇✐t❤ α(L(Kn,n)) = ω(L(Kn,n)) = n✳ Pr♦♣♦s✐t✐♦♥ ▲❡t ✷ ❛♥❞ ❧❡t ❜❡ t❤❡ ❈❛②❧❡② ❣r❛♣❤ ♦♥
✷ ✹ ✇✐t❤
❝♦♥♥❡❝t✐♦♥ s❡t ✵ ✶ ✵ ✸ ✵ ✷ ✷ ✷ ✷ ✶ ✵ ✵ ✶ ✳ ❚❤❡♥ ✐s ❛ ❝♦♥♥❡❝t❡❞ ❈■❙ ❣r❛♣❤ ♦❢ ♦r❞❡r ✽ ❛♥❞ ✈❛❧❡♥❝② ✷ ✸ ✇✐t❤ ✷ ❛♥❞ ✹✳
❆❞❡♠✐r ❍✉❥❞✉r♦✈✐➣ ❱❡rt❡①✲tr❛♥s✐t✐✈❡ ❈■❙ ❣r❛♣❤s
Pr♦♣♦s✐t✐♦♥ ■❢ n ≥ ✶ t❤❡♥ L(Kn,n) ✐s ❛ ❝♦♥♥❡❝t❡❞ ✈❡rt❡①✲tr❛♥s✐t✐✈❡ ❈■❙ ❣r❛♣❤ ♦❢ ♦r❞❡r n✷ ❛♥❞ ✈❛❧❡♥❝② ✷(n − ✶) ✇✐t❤ α(L(Kn,n)) = ω(L(Kn,n)) = n✳ Pr♦♣♦s✐t✐♦♥ ▲❡t n ≥ ✷ ❛♥❞ ❧❡t Rn ❜❡ t❤❡ ❈❛②❧❡② ❣r❛♣❤ ♦♥ Z✷n × Z✹ ✇✐t❤ ❝♦♥♥❡❝t✐♦♥ s❡t S = {(✵, ✶), (✵, ✸), (n, ✵), (n, ✷), (✷i, ✷), (✷i + ✶, ✵) | ✵ ≤ i ≤ n − ✶}✳ ❚❤❡♥ Rn ✐s ❛ ❝♦♥♥❡❝t❡❞ ❈■❙ ❣r❛♣❤ ♦❢ ♦r❞❡r ✽n ❛♥❞ ✈❛❧❡♥❝② ✷n + ✸ ✇✐t❤ α(Rn) = ✷n ❛♥❞ ω(Rn) = ✹✳
❆❞❡♠✐r ❍✉❥❞✉r♦✈✐➣ ❱❡rt❡①✲tr❛♥s✐t✐✈❡ ❈■❙ ❣r❛♣❤s
Pr♦♣♦s✐t✐♦♥ ▲❡t n ≥ ✷ ❛♥❞ ❧❡t Sn ❜❡ t❤❡ ❈❛②❧❡② ❣r❛♣❤ ♦♥ Z✷n × Z✹ ✇✐t❤ ❝♦♥♥❡❝t✐♦♥ s❡t S = {(✵, ✶), (✵, ✸), (✷i +✶, ✵), (✷i +✶, ✶), (✷i +✶, ✸) | ✵ ≤ i ≤ n−✶}✳ ❚❤❡♥ Sn ✐s ❛ ❝♦♥♥❡❝t❡❞ ❈■❙ ❣r❛♣❤ ♦❢ ♦r❞❡r ✽n ❛♥❞ ✈❛❧❡♥❝② ✸n + ✷ ✇✐t❤ α(Sn) = ✷n ❛♥❞ ω(Sn) = ✹✳ ❆❧❧ ♦❢ t❤❡ ❛❜♦✈❡ ❡①❛♠♣❧❡s ❛r❡ ❈❛②❧❡② ❣r❛♣❤s✳ ❉♦❡s t❤❡r❡ ❡①✐st ❛ ✈❡rt❡①✲tr❛♥s✐t✐✈❡ ❈■❙ ❣r❛♣❤ ✇❤✐❝❤ ✐s ♥♦t ❛ ❈❛②❧❡② ❣r❛♣❤❄
❆❞❡♠✐r ❍✉❥❞✉r♦✈✐➣ ❱❡rt❡①✲tr❛♥s✐t✐✈❡ ❈■❙ ❣r❛♣❤s
Pr♦♣♦s✐t✐♦♥ ▲❡t n ≥ ✷ ❛♥❞ ❧❡t Sn ❜❡ t❤❡ ❈❛②❧❡② ❣r❛♣❤ ♦♥ Z✷n × Z✹ ✇✐t❤ ❝♦♥♥❡❝t✐♦♥ s❡t S = {(✵, ✶), (✵, ✸), (✷i +✶, ✵), (✷i +✶, ✶), (✷i +✶, ✸) | ✵ ≤ i ≤ n−✶}✳ ❚❤❡♥ Sn ✐s ❛ ❝♦♥♥❡❝t❡❞ ❈■❙ ❣r❛♣❤ ♦❢ ♦r❞❡r ✽n ❛♥❞ ✈❛❧❡♥❝② ✸n + ✷ ✇✐t❤ α(Sn) = ✷n ❛♥❞ ω(Sn) = ✹✳ ❆❧❧ ♦❢ t❤❡ ❛❜♦✈❡ ❡①❛♠♣❧❡s ❛r❡ ❈❛②❧❡② ❣r❛♣❤s✳ ❉♦❡s t❤❡r❡ ❡①✐st ❛ ✈❡rt❡①✲tr❛♥s✐t✐✈❡ ❈■❙ ❣r❛♣❤ ✇❤✐❝❤ ✐s ♥♦t ❛ ❈❛②❧❡② ❣r❛♣❤❄
❆❞❡♠✐r ❍✉❥❞✉r♦✈✐➣ ❱❡rt❡①✲tr❛♥s✐t✐✈❡ ❈■❙ ❣r❛♣❤s
❉❡✜♥✐t✐♦♥ ❋♦r n ≥ ✸✱ ❧❡t PX(n) ❜❡ t❤❡ ❣r❛♣❤ ✇✐t❤ ✈❡rt❡①✲s❡t Zn × Z✷ × Z✷ ❛♥❞ ❡❞❣❡✲s❡t {(i, x, y), (i + ✶, y, z) | i ∈ Zn, x, y, z ∈ Z✷}✳ ❚♦ PX(n)✱ ✇❡ ❛❞❞ t❤❡ ❢♦❧❧♦✇✐♥❣ s❡t ♦❢ ❡❞❣❡s {(i, x, y), (i, u, v) | i ∈ Zn, x, y, u, v ∈ Z✷, (x, y) = (u, v)} t♦ ♦❜t❛✐♥ t❤❡ ❣r❛♣❤ Qn✳ Pr♦♣♦s✐t✐♦♥ ▲❡t ✹✳ ❚❤❡ ❣r❛♣❤ ✐s ❛ ❝♦♥♥❡❝t❡❞ ✈❡rt❡①✲tr❛♥s✐t✐✈❡ ❈■❙ ❣r❛♣❤ ♦❢ ♦r❞❡r ✹ ❛♥❞ ✈❛❧❡♥❝② ✼ ✇✐t❤ ❛♥❞ ✹✳ ▼♦r❡♦✈❡r✱ ✐❢ ✐s ♣r✐♠❡ t❤❡♥ ✐s ♥♦t ❛ ❈❛②❧❡② ❣r❛♣❤✳
❆❞❡♠✐r ❍✉❥❞✉r♦✈✐➣ ❱❡rt❡①✲tr❛♥s✐t✐✈❡ ❈■❙ ❣r❛♣❤s
❉❡✜♥✐t✐♦♥ ❋♦r n ≥ ✸✱ ❧❡t PX(n) ❜❡ t❤❡ ❣r❛♣❤ ✇✐t❤ ✈❡rt❡①✲s❡t Zn × Z✷ × Z✷ ❛♥❞ ❡❞❣❡✲s❡t {(i, x, y), (i + ✶, y, z) | i ∈ Zn, x, y, z ∈ Z✷}✳ ❚♦ PX(n)✱ ✇❡ ❛❞❞ t❤❡ ❢♦❧❧♦✇✐♥❣ s❡t ♦❢ ❡❞❣❡s {(i, x, y), (i, u, v) | i ∈ Zn, x, y, u, v ∈ Z✷, (x, y) = (u, v)} t♦ ♦❜t❛✐♥ t❤❡ ❣r❛♣❤ Qn✳ Pr♦♣♦s✐t✐♦♥ ▲❡t n ≥ ✹✳ ❚❤❡ ❣r❛♣❤ Qn ✐s ❛ ❝♦♥♥❡❝t❡❞ ✈❡rt❡①✲tr❛♥s✐t✐✈❡ ❈■❙ ❣r❛♣❤ ♦❢ ♦r❞❡r ✹n ❛♥❞ ✈❛❧❡♥❝② ✼ ✇✐t❤ α(Qn) = n ❛♥❞ ω(Qn) = ✹✳ ▼♦r❡♦✈❡r✱ ✐❢ n ✐s ♣r✐♠❡ t❤❡♥ Qn ✐s ♥♦t ❛ ❈❛②❧❡② ❣r❛♣❤✳
❆❞❡♠✐r ❍✉❥❞✉r♦✈✐➣ ❱❡rt❡①✲tr❛♥s✐t✐✈❡ ❈■❙ ❣r❛♣❤s
❯s✐♥❣ ●♦r❞♦♥ ❘♦②❧❡✬s t❛❜❧❡ ♦❢ ✈❡rt❡①✲tr❛♥s✐t✐✈❡ ❣r❛♣❤s ♦❢ ♦r❞❡r ❛t ♠♦st ✸✷✱ ✇❡ ♦❜t❛✐♥ t❤❡ ❢♦❧❧♦✇✐♥❣ ✇✐t❤ t❤❡ ❤❡❧♣ ♦❢ ❛ ❝♦♠♣✉t❡r✳ Pr♦♣♦s✐t✐♦♥ ▲❡t F ❜❡ t❤❡ ❢❛♠✐❧② ❝♦♥t❛✐♥✐♥❣ t❤❡ ❢♦❧❧♦✇✐♥❣ ❣r❛♣❤s✿
✶ Kn✱ n ≥ ✶✱ ✷ L(Kn,n)✱ n ≥ ✸✱ ✸ Qn✱ n ≥ ✹✱ ✹ Rn✱ n ≥ ✸✱ ✺ Sn✱ n ≥ ✷✱
❛♥❞ ❧❡t F ❜❡ t❤❡ ❝❧♦s✉r❡ ♦❢ F ✉♥❞❡r t❤❡ ♦♣❡r❛t✐♦♥s ♦❢ t❛❦✐♥❣ ❝♦♠♣❧❡♠❡♥ts ❛♥❞ ❧❡①✐❝♦❣r❛♣❤✐❝ ♣r♦❞✉❝ts✳ ❚❤❡♥✱ ✉♣ t♦ ✐s♦♠♦r♣❤✐s♠✱ ❡✈❡r② ✈❡rt❡①✲tr❛♥s✐t✐✈❡ ❈■❙ ❣r❛♣❤ ♦❢ ♦r❞❡r ❛t ♠♦st ✸✷ ✐s ✐♥ F✳
❆❞❡♠✐r ❍✉❥❞✉r♦✈✐➣ ❱❡rt❡①✲tr❛♥s✐t✐✈❡ ❈■❙ ❣r❛♣❤s
❚❤❡♦r❡♠ ▲❡t ❜❡ ❛ ❝♦♥♥❡❝t❡❞ ❈■❙ ❣r❛♣❤ ✇✐t❤ ✷✳ ❚❤❡♥ ✐s ❝♦♠♣❧❡t❡ ❜✐♣❛rt✐t❡✳ ❈♦r♦❧❧❛r② ▲❡t ❜❡ ❛ ❝♦♥♥❡❝t❡❞ ✈❡rt❡①✲tr❛♥s✐t✐✈❡ ❈■❙ ❣r❛♣❤ ✇✐t❤ ✷✳ ❚❤❡♥ ✱ ❢♦r s♦♠❡ ♣♦s✐t✐✈❡ ✐♥t❡❣❡r ✳ ❚❤❡♦r❡♠ ▲❡t ❜❡ ❛ ❝♦♥♥❡❝t❡❞ ✈❡rt❡①✲tr❛♥s✐t✐✈❡ ❣r❛♣❤ ✇✐t❤ ✸✳ ❚❤❡♥ ✐s ❛ ❈■❙ ❣r❛♣❤ ✐❢ ❛♥❞ ♦♥❧② ✐❢
✶
✸ ✶ ✱ ♦r
✷
✸ ✸ ✶
❢♦r s♦♠❡ ♣♦s✐t✐✈❡ ✐♥t❡❣❡r ✳
❆❞❡♠✐r ❍✉❥❞✉r♦✈✐➣ ❱❡rt❡①✲tr❛♥s✐t✐✈❡ ❈■❙ ❣r❛♣❤s
❚❤❡♦r❡♠ ▲❡t Γ ❜❡ ❛ ❝♦♥♥❡❝t❡❞ ❈■❙ ❣r❛♣❤ ✇✐t❤ ω(Γ) = ✷✳ ❚❤❡♥ Γ ✐s ❝♦♠♣❧❡t❡ ❜✐♣❛rt✐t❡✳ ❈♦r♦❧❧❛r② ▲❡t ❜❡ ❛ ❝♦♥♥❡❝t❡❞ ✈❡rt❡①✲tr❛♥s✐t✐✈❡ ❈■❙ ❣r❛♣❤ ✇✐t❤ ✷✳ ❚❤❡♥ ✱ ❢♦r s♦♠❡ ♣♦s✐t✐✈❡ ✐♥t❡❣❡r ✳ ❚❤❡♦r❡♠ ▲❡t ❜❡ ❛ ❝♦♥♥❡❝t❡❞ ✈❡rt❡①✲tr❛♥s✐t✐✈❡ ❣r❛♣❤ ✇✐t❤ ✸✳ ❚❤❡♥ ✐s ❛ ❈■❙ ❣r❛♣❤ ✐❢ ❛♥❞ ♦♥❧② ✐❢
✶
✸ ✶ ✱ ♦r
✷
✸ ✸ ✶
❢♦r s♦♠❡ ♣♦s✐t✐✈❡ ✐♥t❡❣❡r ✳
❆❞❡♠✐r ❍✉❥❞✉r♦✈✐➣ ❱❡rt❡①✲tr❛♥s✐t✐✈❡ ❈■❙ ❣r❛♣❤s
❚❤❡♦r❡♠ ▲❡t Γ ❜❡ ❛ ❝♦♥♥❡❝t❡❞ ❈■❙ ❣r❛♣❤ ✇✐t❤ ω(Γ) = ✷✳ ❚❤❡♥ Γ ✐s ❝♦♠♣❧❡t❡ ❜✐♣❛rt✐t❡✳ ❈♦r♦❧❧❛r② ▲❡t Γ ❜❡ ❛ ❝♦♥♥❡❝t❡❞ ✈❡rt❡①✲tr❛♥s✐t✐✈❡ ❈■❙ ❣r❛♣❤ ✇✐t❤ ω(Γ) = ✷✳ ❚❤❡♥ Γ ∼ = Kn,n✱ ❢♦r s♦♠❡ ♣♦s✐t✐✈❡ ✐♥t❡❣❡r n✳ ❚❤❡♦r❡♠ ▲❡t ❜❡ ❛ ❝♦♥♥❡❝t❡❞ ✈❡rt❡①✲tr❛♥s✐t✐✈❡ ❣r❛♣❤ ✇✐t❤ ✸✳ ❚❤❡♥ ✐s ❛ ❈■❙ ❣r❛♣❤ ✐❢ ❛♥❞ ♦♥❧② ✐❢
✶
✸ ✶ ✱ ♦r
✷
✸ ✸ ✶
❢♦r s♦♠❡ ♣♦s✐t✐✈❡ ✐♥t❡❣❡r ✳
❆❞❡♠✐r ❍✉❥❞✉r♦✈✐➣ ❱❡rt❡①✲tr❛♥s✐t✐✈❡ ❈■❙ ❣r❛♣❤s
❚❤❡♦r❡♠ ▲❡t Γ ❜❡ ❛ ❝♦♥♥❡❝t❡❞ ❈■❙ ❣r❛♣❤ ✇✐t❤ ω(Γ) = ✷✳ ❚❤❡♥ Γ ✐s ❝♦♠♣❧❡t❡ ❜✐♣❛rt✐t❡✳ ❈♦r♦❧❧❛r② ▲❡t Γ ❜❡ ❛ ❝♦♥♥❡❝t❡❞ ✈❡rt❡①✲tr❛♥s✐t✐✈❡ ❈■❙ ❣r❛♣❤ ✇✐t❤ ω(Γ) = ✷✳ ❚❤❡♥ Γ ∼ = Kn,n✱ ❢♦r s♦♠❡ ♣♦s✐t✐✈❡ ✐♥t❡❣❡r n✳ ❚❤❡♦r❡♠ ▲❡t Γ ❜❡ ❛ ❝♦♥♥❡❝t❡❞ ✈❡rt❡①✲tr❛♥s✐t✐✈❡ ❣r❛♣❤ ✇✐t❤ ω(Γ) = ✸✳ ❚❤❡♥ Γ ✐s ❛ ❈■❙ ❣r❛♣❤ ✐❢ ❛♥❞ ♦♥❧② ✐❢
✶ Γ ∼
= K✸[nK✶]✱ ♦r
✷ Γ ∼
= L(K✸,✸)[nK✶] ❢♦r s♦♠❡ ♣♦s✐t✐✈❡ ✐♥t❡❣❡r n✳
❆❞❡♠✐r ❍✉❥❞✉r♦✈✐➣ ❱❡rt❡①✲tr❛♥s✐t✐✈❡ ❈■❙ ❣r❛♣❤s
❚❤❡♦r❡♠ ✭❑♦st♦❝❤❦❛✱ ✶✾✽✵✮ ▲❡t Γ ❜❡ ❛ ❣r❛♣❤ ❛♥❞ ❧❡t Q ❜❡ ❛ ❝♦❧❧❡❝t✐♦♥ ♦❢ ♠❛①✐♠✉♠ ❝❧✐q✉❡s ♦❢ Γ✳ ▲❡t ΓQ ❜❡ t❤❡ ❣r❛♣❤ ♦❢ ♠❛①✐♠❛❧ ❝❧✐q✉❡s ♦❢ Γ✱ t❤❛t ✐s✱ t❤❡ ❣r❛♣❤ ✇✐t❤ ♠❛①✐♠✉♠ ❝❧✐q✉❡s ♦❢ Γ ❛s ✈❡rt✐❝❡s✱ ❛♥❞ t✇♦ s✉❝❤ ❝❧✐q✉❡s ❛r❡ ❛❞❥❛❝❡♥t ✐♥ ΓQ ✐❢ t❤❡② ✐♥t❡rs❡❝t ✐♥ Γ✳ ■❢ ω(Γ) > ✷
✸(∆(Γ) + ✶) ❛♥❞ ✐❢ ΓQ ✐s ❝♦♥♥❡❝t❡❞ t❤❡♥ ∩Q = ∅✳
▲❡♠♠❛ ▲❡t ❜❡ ❛ ❝♦♥♥❡❝t❡❞✱ ✲r❡❣✉❧❛r✱ ✇❡❧❧✲❝♦✈❡r❡❞✱ ❝♦✲✇❡❧❧✲❝♦✈❡r❡❞ ❣r❛♣❤✳ ❚❤❡♥ ❡✐t❤❡r
✷ ✸
✶ ♦r ✐s ❛ ❝♦♠♣❧❡t❡ ❣r❛♣❤✳ ■♥ ♣❛rt✐❝✉❧❛r✱ ✐❢ ✐s ♥♦t ❝♦♠♣❧❡t❡ ❣r❛♣❤ t❤❡♥✿ ✐❢ ✸ t❤❡♥ ✷ ✐❢ ✹ t❤❡♥ ✸ ✐❢ ✺ ♦r ✻ t❤❡♥ ✹ ✐❢ ✼ t❤❡♥ ✺✳
❆❞❡♠✐r ❍✉❥❞✉r♦✈✐➣ ❱❡rt❡①✲tr❛♥s✐t✐✈❡ ❈■❙ ❣r❛♣❤s
❚❤❡♦r❡♠ ✭❑♦st♦❝❤❦❛✱ ✶✾✽✵✮ ▲❡t Γ ❜❡ ❛ ❣r❛♣❤ ❛♥❞ ❧❡t Q ❜❡ ❛ ❝♦❧❧❡❝t✐♦♥ ♦❢ ♠❛①✐♠✉♠ ❝❧✐q✉❡s ♦❢ Γ✳ ▲❡t ΓQ ❜❡ t❤❡ ❣r❛♣❤ ♦❢ ♠❛①✐♠❛❧ ❝❧✐q✉❡s ♦❢ Γ✱ t❤❛t ✐s✱ t❤❡ ❣r❛♣❤ ✇✐t❤ ♠❛①✐♠✉♠ ❝❧✐q✉❡s ♦❢ Γ ❛s ✈❡rt✐❝❡s✱ ❛♥❞ t✇♦ s✉❝❤ ❝❧✐q✉❡s ❛r❡ ❛❞❥❛❝❡♥t ✐♥ ΓQ ✐❢ t❤❡② ✐♥t❡rs❡❝t ✐♥ Γ✳ ■❢ ω(Γ) > ✷
✸(∆(Γ) + ✶) ❛♥❞ ✐❢ ΓQ ✐s ❝♦♥♥❡❝t❡❞ t❤❡♥ ∩Q = ∅✳
▲❡♠♠❛ ▲❡t Γ ❜❡ ❛ ❝♦♥♥❡❝t❡❞✱ k✲r❡❣✉❧❛r✱ ✇❡❧❧✲❝♦✈❡r❡❞✱ ❝♦✲✇❡❧❧✲❝♦✈❡r❡❞ ❣r❛♣❤✳ ❚❤❡♥ ❡✐t❤❡r ω(Γ) ≤ ✷
✸(k + ✶) ♦r Γ ✐s ❛ ❝♦♠♣❧❡t❡ ❣r❛♣❤✳
■♥ ♣❛rt✐❝✉❧❛r✱ ✐❢ ✐s ♥♦t ❝♦♠♣❧❡t❡ ❣r❛♣❤ t❤❡♥✿ ✐❢ ✸ t❤❡♥ ✷ ✐❢ ✹ t❤❡♥ ✸ ✐❢ ✺ ♦r ✻ t❤❡♥ ✹ ✐❢ ✼ t❤❡♥ ✺✳
❆❞❡♠✐r ❍✉❥❞✉r♦✈✐➣ ❱❡rt❡①✲tr❛♥s✐t✐✈❡ ❈■❙ ❣r❛♣❤s
❚❤❡♦r❡♠ ✭❑♦st♦❝❤❦❛✱ ✶✾✽✵✮ ▲❡t Γ ❜❡ ❛ ❣r❛♣❤ ❛♥❞ ❧❡t Q ❜❡ ❛ ❝♦❧❧❡❝t✐♦♥ ♦❢ ♠❛①✐♠✉♠ ❝❧✐q✉❡s ♦❢ Γ✳ ▲❡t ΓQ ❜❡ t❤❡ ❣r❛♣❤ ♦❢ ♠❛①✐♠❛❧ ❝❧✐q✉❡s ♦❢ Γ✱ t❤❛t ✐s✱ t❤❡ ❣r❛♣❤ ✇✐t❤ ♠❛①✐♠✉♠ ❝❧✐q✉❡s ♦❢ Γ ❛s ✈❡rt✐❝❡s✱ ❛♥❞ t✇♦ s✉❝❤ ❝❧✐q✉❡s ❛r❡ ❛❞❥❛❝❡♥t ✐♥ ΓQ ✐❢ t❤❡② ✐♥t❡rs❡❝t ✐♥ Γ✳ ■❢ ω(Γ) > ✷
✸(∆(Γ) + ✶) ❛♥❞ ✐❢ ΓQ ✐s ❝♦♥♥❡❝t❡❞ t❤❡♥ ∩Q = ∅✳
▲❡♠♠❛ ▲❡t Γ ❜❡ ❛ ❝♦♥♥❡❝t❡❞✱ k✲r❡❣✉❧❛r✱ ✇❡❧❧✲❝♦✈❡r❡❞✱ ❝♦✲✇❡❧❧✲❝♦✈❡r❡❞ ❣r❛♣❤✳ ❚❤❡♥ ❡✐t❤❡r ω(Γ) ≤ ✷
✸(k + ✶) ♦r Γ ✐s ❛ ❝♦♠♣❧❡t❡ ❣r❛♣❤✳
■♥ ♣❛rt✐❝✉❧❛r✱ ✐❢ Γ ✐s ♥♦t ❝♦♠♣❧❡t❡ ❣r❛♣❤ t❤❡♥✿ ✐❢ k = ✸ t❤❡♥ ω(Γ) ≤ ✷ ✐❢ k = ✹ t❤❡♥ ω(Γ) ≤ ✸ ✐❢ k = ✺ ♦r ✻ t❤❡♥ ω(Γ) ≤ ✹ ✐❢ k = ✼ t❤❡♥ ω(Γ) ≤ ✺✳
❆❞❡♠✐r ❍✉❥❞✉r♦✈✐➣ ❱❡rt❡①✲tr❛♥s✐t✐✈❡ ❈■❙ ❣r❛♣❤s
❚❤❡♦r❡♠ ▲❡t Γ ❜❡ ❛ ❝♦♥♥❡❝t❡❞ ✈❡rt❡①✲tr❛♥s✐t✐✈❡ ❣r❛♣❤ ♦❢ ✈❛❧❡♥❝② k ≤ ✼✳ ❚❤❡♥ Γ ✐s ❛ ❈■❙ ❣r❛♣❤ ✐❢ ❛♥❞ ♦♥❧② ✐❢ Γ ✐s ✐s♦♠♦r♣❤✐❝ t♦ ♦♥❡ ♦❢ t❤❡ ❢♦❧❧♦✇✐♥❣ ❣r❛♣❤s✿ ✭k = ✶✮ K✷❀ ✭k = ✷✮ K✸✱ K✷,✷ ✭k = ✸✮ K✹✱ K✸,✸ ✭k = ✹✮ K✺✱ K✹,✹✱ L(K✸,✸)✱ K✸[✷K✶] ✭k = ✺✮ K✻✱ K✺,✺✱ C✹[K✷]✱ ✭k = ✻✮ K✼✱ K✻,✻✱ L(K✹,✹)✱ K✸[✸K✶]✱ K✹[✷K✶] ✭k = ✼✮ K✽✱ K✼,✼✱ K✸,✸[K✷]✱ Qn ❢♦r n ≥ ✹✳
❆❞❡♠✐r ❍✉❥❞✉r♦✈✐➣ ❱❡rt❡①✲tr❛♥s✐t✐✈❡ ❈■❙ ❣r❛♣❤s
■❢ k ≤ ✹ t❤❡ r❡s✉❧t ❢♦❧❧♦✇s ❢r♦♠ t❤❡ r❡s✉❧t ❢♦r ω = ✷ ❛♥❞ ω = ✸✳ ■❢ ✺ ❛♥❞ ✹✱ ❝♦♥s✐❞❡r ✜rst t❤❡ ❝❛s❡ ✇❤❡♥ ✐s ✐rr❡❞✉❝✐❜❧❡✳ ❚❤❡ ❧♦❝❛❧ ❣r❛♣❤ ♦❢ ❤❛s ✺ ✈❡rt✐❝❡s✱ ♠❛①✐♠❛❧ ❝❧✐q✉❡s ♦❢ s✐③❡ ✸ ✇✐t❤ ♥♦ t✇♦ tr✐❛♥❣❧❡s ❤❛✈✐♥❣ ❛♥ ❡❞❣❡ ✐♥ ❝♦♠♠♦♥✳ ❚❤❡r❡ ✐s ♦♥❧② ♦♥❡ s✉❝❤ ❣r❛♣❤✳
❋✐❣✉r❡✿ ❚❤❡ ❣r❛♣❤
✷
■t ✐s ♥♦✇ ❡❛s② t♦ ❝♦♥❝❧✉❞❡ t❤❛t ✐s t❤❡ ❧❡①✐❝♦❣r❛♣❤✐❝ ♣r♦❞✉❝t ♦❢ ❛ ❝②❝❧❡ ✭ ✹ ✇✐t❤
✷✳ ❚❤❡♥
✹ ❛♥❞
✹ ✷ ✳
❆❞❡♠✐r ❍✉❥❞✉r♦✈✐➣ ❱❡rt❡①✲tr❛♥s✐t✐✈❡ ❈■❙ ❣r❛♣❤s
■❢ k ≤ ✹ t❤❡ r❡s✉❧t ❢♦❧❧♦✇s ❢r♦♠ t❤❡ r❡s✉❧t ❢♦r ω = ✷ ❛♥❞ ω = ✸✳ ■❢ k = ✺ ❛♥❞ ω(Γ) = ✹✱ ❝♦♥s✐❞❡r ✜rst t❤❡ ❝❛s❡ ✇❤❡♥ Γ ✐s ✐rr❡❞✉❝✐❜❧❡✳ ❚❤❡ ❧♦❝❛❧ ❣r❛♣❤ ♦❢ Γ ❤❛s ✺ ✈❡rt✐❝❡s✱ ♠❛①✐♠❛❧ ❝❧✐q✉❡s ♦❢ s✐③❡ ✸ ✇✐t❤ ♥♦ t✇♦ tr✐❛♥❣❧❡s ❤❛✈✐♥❣ ❛♥ ❡❞❣❡ ✐♥ ❝♦♠♠♦♥✳ ❚❤❡r❡ ✐s ♦♥❧② ♦♥❡ s✉❝❤ ❣r❛♣❤✳
❋✐❣✉r❡✿ ❚❤❡ ❣r❛♣❤
✷
■t ✐s ♥♦✇ ❡❛s② t♦ ❝♦♥❝❧✉❞❡ t❤❛t ✐s t❤❡ ❧❡①✐❝♦❣r❛♣❤✐❝ ♣r♦❞✉❝t ♦❢ ❛ ❝②❝❧❡ ✭ ✹ ✇✐t❤
✷✳ ❚❤❡♥
✹ ❛♥❞
✹ ✷ ✳
❆❞❡♠✐r ❍✉❥❞✉r♦✈✐➣ ❱❡rt❡①✲tr❛♥s✐t✐✈❡ ❈■❙ ❣r❛♣❤s
■❢ k ≤ ✹ t❤❡ r❡s✉❧t ❢♦❧❧♦✇s ❢r♦♠ t❤❡ r❡s✉❧t ❢♦r ω = ✷ ❛♥❞ ω = ✸✳ ■❢ k = ✺ ❛♥❞ ω(Γ) = ✹✱ ❝♦♥s✐❞❡r ✜rst t❤❡ ❝❛s❡ ✇❤❡♥ Γ ✐s ✐rr❡❞✉❝✐❜❧❡✳ ❚❤❡ ❧♦❝❛❧ ❣r❛♣❤ ♦❢ Γ ❤❛s ✺ ✈❡rt✐❝❡s✱ ♠❛①✐♠❛❧ ❝❧✐q✉❡s ♦❢ s✐③❡ ✸ ✇✐t❤ ♥♦ t✇♦ tr✐❛♥❣❧❡s ❤❛✈✐♥❣ ❛♥ ❡❞❣❡ ✐♥ ❝♦♠♠♦♥✳ ❚❤❡r❡ ✐s ♦♥❧② ♦♥❡ s✉❝❤ ❣r❛♣❤✳
❋✐❣✉r❡✿ ❚❤❡ ❣r❛♣❤ T✷
■t ✐s ♥♦✇ ❡❛s② t♦ ❝♦♥❝❧✉❞❡ t❤❛t Γ ✐s t❤❡ ❧❡①✐❝♦❣r❛♣❤✐❝ ♣r♦❞✉❝t ♦❢ ❛ ❝②❝❧❡ Cn ✭n ≥ ✹) ✇✐t❤ K✷✳ ❚❤❡♥ n = ✹ ❛♥❞ Γ = C✹[K✷]✳
❆❞❡♠✐r ❍✉❥❞✉r♦✈✐➣ ❱❡rt❡①✲tr❛♥s✐t✐✈❡ ❈■❙ ❣r❛♣❤s
❚❤❡ ❝❛s❡s ✇❤❡♥ (k, ω) ∈ {(✻, ✹), (✼, ✺)} ❛r❡ s♦❧✈❡❞ s✐♠✐❧❛r❧②✳ ■❢ ✼ ❛♥❞ ✹ t❤❡♥ t❤❡r❡ ❛r❡ t✇♦ ♣♦ss✐❜❧❡ ❧♦❝❛❧ ❣r❛♣❤s✿ ❚❤❡ ✜rst ❧♦❝❛❧ ❣r❛♣❤ ✐s ❡❛s② t♦ ❤❛♥❞❧❡✱ ✇❡ ❤❛✈❡
✸ ✸ ✷ ✳
❚❤❡ ❝❛s❡ ♦❢ t❤❡ s❡❝♦♥❞ ❧♦❝❛❧ ❣r❛♣❤ ✐s ♠✉❝❤ ♠♦r❡ ❞✐✣❝✉❧t✱ ❛♥❞ ✐t r❡s✉❧ts ✐♥ t❤❡ ✐♥✜♥✐t❡ ❢❛♠✐❧② ✳
❆❞❡♠✐r ❍✉❥❞✉r♦✈✐➣ ❱❡rt❡①✲tr❛♥s✐t✐✈❡ ❈■❙ ❣r❛♣❤s
❚❤❡ ❝❛s❡s ✇❤❡♥ (k, ω) ∈ {(✻, ✹), (✼, ✺)} ❛r❡ s♦❧✈❡❞ s✐♠✐❧❛r❧②✳ ■❢ k = ✼ ❛♥❞ ω(Γ) = ✹ t❤❡♥ t❤❡r❡ ❛r❡ t✇♦ ♣♦ss✐❜❧❡ ❧♦❝❛❧ ❣r❛♣❤s✿ ❚❤❡ ✜rst ❧♦❝❛❧ ❣r❛♣❤ ✐s ❡❛s② t♦ ❤❛♥❞❧❡✱ ✇❡ ❤❛✈❡
✸ ✸ ✷ ✳
❚❤❡ ❝❛s❡ ♦❢ t❤❡ s❡❝♦♥❞ ❧♦❝❛❧ ❣r❛♣❤ ✐s ♠✉❝❤ ♠♦r❡ ❞✐✣❝✉❧t✱ ❛♥❞ ✐t r❡s✉❧ts ✐♥ t❤❡ ✐♥✜♥✐t❡ ❢❛♠✐❧② ✳
❆❞❡♠✐r ❍✉❥❞✉r♦✈✐➣ ❱❡rt❡①✲tr❛♥s✐t✐✈❡ ❈■❙ ❣r❛♣❤s
❚❤❡ ❝❛s❡s ✇❤❡♥ (k, ω) ∈ {(✻, ✹), (✼, ✺)} ❛r❡ s♦❧✈❡❞ s✐♠✐❧❛r❧②✳ ■❢ k = ✼ ❛♥❞ ω(Γ) = ✹ t❤❡♥ t❤❡r❡ ❛r❡ t✇♦ ♣♦ss✐❜❧❡ ❧♦❝❛❧ ❣r❛♣❤s✿ ❚❤❡ ✜rst ❧♦❝❛❧ ❣r❛♣❤ ✐s ❡❛s② t♦ ❤❛♥❞❧❡✱ ✇❡ ❤❛✈❡ Γ = K✸,✸[K✷]✳ ❚❤❡ ❝❛s❡ ♦❢ t❤❡ s❡❝♦♥❞ ❧♦❝❛❧ ❣r❛♣❤ ✐s ♠✉❝❤ ♠♦r❡ ❞✐✣❝✉❧t✱ ❛♥❞ ✐t r❡s✉❧ts ✐♥ t❤❡ ✐♥✜♥✐t❡ ❢❛♠✐❧② ✳
❆❞❡♠✐r ❍✉❥❞✉r♦✈✐➣ ❱❡rt❡①✲tr❛♥s✐t✐✈❡ ❈■❙ ❣r❛♣❤s
❚❤❡ ❝❛s❡s ✇❤❡♥ (k, ω) ∈ {(✻, ✹), (✼, ✺)} ❛r❡ s♦❧✈❡❞ s✐♠✐❧❛r❧②✳ ■❢ k = ✼ ❛♥❞ ω(Γ) = ✹ t❤❡♥ t❤❡r❡ ❛r❡ t✇♦ ♣♦ss✐❜❧❡ ❧♦❝❛❧ ❣r❛♣❤s✿ ❚❤❡ ✜rst ❧♦❝❛❧ ❣r❛♣❤ ✐s ❡❛s② t♦ ❤❛♥❞❧❡✱ ✇❡ ❤❛✈❡ Γ = K✸,✸[K✷]✳ ❚❤❡ ❝❛s❡ ♦❢ t❤❡ s❡❝♦♥❞ ❧♦❝❛❧ ❣r❛♣❤ ✐s ♠✉❝❤ ♠♦r❡ ❞✐✣❝✉❧t✱ ❛♥❞ ✐t r❡s✉❧ts ✐♥ t❤❡ ✐♥✜♥✐t❡ ❢❛♠✐❧② Qn✳
❆❞❡♠✐r ❍✉❥❞✉r♦✈✐➣ ❱❡rt❡①✲tr❛♥s✐t✐✈❡ ❈■❙ ❣r❛♣❤s
◗✉❡st✐♦♥ ❉♦❡s ❡✈❡r② ❈■❙ ❣r❛♣❤ Γ s❛t✐s❢② α(Γ)ω(Γ) ≥ |V (Γ)|❄ ◗✉❡st✐♦♥ ❉♦❡s t❤❡ ❝❤r♦♠❛t✐❝ ♥✉♠❜❡r ♦❢ ❡✈❡r② ✈❡rt❡① tr❛♥s✐t✐✈❡ ❈■❙ ❣r❛♣❤ ❡q✉❛❧ ✐ts ❝❧✐q✉❡ ♥✉♠❜❡r❄ ◗✉❡st✐♦♥ ■s ❡✈❡r② ❝♦♥♥❡❝t❡❞ r❡❣✉❧❛r ✐rr❡❞✉❝✐❜❧❡ ✇❡❧❧✲❝♦✈❡r❡❞ ❝♦✲✇❡❧❧✲❝♦✈❡r❡❞ ❈■❙ ❣r❛♣❤ ✈❡rt❡①✲tr❛♥s✐t✐✈❡❄
❆❞❡♠✐r ❍✉❥❞✉r♦✈✐➣ ❱❡rt❡①✲tr❛♥s✐t✐✈❡ ❈■❙ ❣r❛♣❤s
◗✉❡st✐♦♥ ❉♦❡s ❡✈❡r② ❈■❙ ❣r❛♣❤ Γ s❛t✐s❢② α(Γ)ω(Γ) ≥ |V (Γ)|❄ ◗✉❡st✐♦♥ ❉♦❡s t❤❡ ❝❤r♦♠❛t✐❝ ♥✉♠❜❡r ♦❢ ❡✈❡r② ✈❡rt❡① tr❛♥s✐t✐✈❡ ❈■❙ ❣r❛♣❤ ❡q✉❛❧ ✐ts ❝❧✐q✉❡ ♥✉♠❜❡r❄ ◗✉❡st✐♦♥ ■s ❡✈❡r② ❝♦♥♥❡❝t❡❞ r❡❣✉❧❛r ✐rr❡❞✉❝✐❜❧❡ ✇❡❧❧✲❝♦✈❡r❡❞ ❝♦✲✇❡❧❧✲❝♦✈❡r❡❞ ❈■❙ ❣r❛♣❤ ✈❡rt❡①✲tr❛♥s✐t✐✈❡❄
❆❞❡♠✐r ❍✉❥❞✉r♦✈✐➣ ❱❡rt❡①✲tr❛♥s✐t✐✈❡ ❈■❙ ❣r❛♣❤s
◗✉❡st✐♦♥ ❉♦❡s ❡✈❡r② ❈■❙ ❣r❛♣❤ Γ s❛t✐s❢② α(Γ)ω(Γ) ≥ |V (Γ)|❄ ◗✉❡st✐♦♥ ❉♦❡s t❤❡ ❝❤r♦♠❛t✐❝ ♥✉♠❜❡r ♦❢ ❡✈❡r② ✈❡rt❡① tr❛♥s✐t✐✈❡ ❈■❙ ❣r❛♣❤ ❡q✉❛❧ ✐ts ❝❧✐q✉❡ ♥✉♠❜❡r❄ ◗✉❡st✐♦♥ ■s ❡✈❡r② ❝♦♥♥❡❝t❡❞ r❡❣✉❧❛r ✐rr❡❞✉❝✐❜❧❡ ✇❡❧❧✲❝♦✈❡r❡❞ ❝♦✲✇❡❧❧✲❝♦✈❡r❡❞ ❈■❙ ❣r❛♣❤ ✈❡rt❡①✲tr❛♥s✐t✐✈❡❄
❆❞❡♠✐r ❍✉❥❞✉r♦✈✐➣ ❱❡rt❡①✲tr❛♥s✐t✐✈❡ ❈■❙ ❣r❛♣❤s
❆❞❡♠✐r ❍✉❥❞✉r♦✈✐➣ ❱❡rt❡①✲tr❛♥s✐t✐✈❡ ❈■❙ ❣r❛♣❤s