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❖♥❧✐♥❡ ▼❛①✲❊❞❣❡✲❈♦❧♦r✐♥❣ ♦❢ P❛t❤s ❛♥❞ ❚r❡❡s ▲❡♥❡ ▼✳ ❋❛✈r❤♦❧❞t ❛♥❞ ❏❡s♣❡r ❲✳ ▼✐❦❦❡❧s❡♥ ❉❡♣❛rt♠❡♥t ♦❢ ▼❛t❤❡♠❛t✐❝s ❛♥❞ ❈♦♠♣✉t❡r ❙❝✐❡♥❝❡ ❯♥✐✈❡rs✐t② ♦❢ ❙♦✉t❤❡r♥ ❉❡♥♠❛r❦ ❚❖▲❆✱ ❏✉❧② ✼✱ ✷✵✶✹ ❏❡s♣❡r ❲✳ ▼✐❦❦❡❧s❡♥ ✶✴✸✹

❊❞❣❡ ❈♦❧♦r✐♥❣ ❆♥ ❡❞❣❡ ❝♦❧♦r✐♥❣ ♦❢ t❤❡ P❡t❡rs❡♥ ❣r❛♣❤ ✉s✐♥❣ 4 ❝♦❧♦rs✳ ❏❡s♣❡r ❲✳ ▼✐❦❦❡❧s❡♥ ✷✴✸✹

❱✐③✐♥❣✬s ❚❤❡♦r❡♠ ▲❡t ❜❡ ❛ s✐♠♣❧❡ ❣r❛♣❤ ♦❢ ♠❛①✐♠✉♠ ❞❡❣r❡❡ ✳ ❚❤❡ ♠✐♥✐♠✉♠ ♥✉♠❜❡r ♦❢ ❝♦❧♦rs ♥❡❡❞❡❞ t♦ ❝♦❧♦r ❛❧❧ ❡❞❣❡s ♦❢ ✐s ❡✐t❤❡r ♦r ✳ ▼✐♥✐♠✉♠ ❊❞❣❡ ❈♦❧♦r✐♥❣ ❈❧❛ss✐❝❛❧ ❡❞❣❡ ❝♦❧♦r✐♥❣✿ ◮ ❈♦❧♦r t❤❡ ❡❞❣❡s ♦❢ ❛ ❣r❛♣❤ ✉s✐♥❣ ❛s ❢❡✇ ❝♦❧♦rs ❛s ♣♦ss✐❜❧❡✳ ❏❡s♣❡r ❲✳ ▼✐❦❦❡❧s❡♥ ✸✴✸✹

▼✐♥✐♠✉♠ ❊❞❣❡ ❈♦❧♦r✐♥❣ ❈❧❛ss✐❝❛❧ ❡❞❣❡ ❝♦❧♦r✐♥❣✿ ◮ ❈♦❧♦r t❤❡ ❡❞❣❡s ♦❢ ❛ ❣r❛♣❤ ✉s✐♥❣ ❛s ❢❡✇ ❝♦❧♦rs ❛s ♣♦ss✐❜❧❡✳ ❱✐③✐♥❣✬s ❚❤❡♦r❡♠ ▲❡t G ❜❡ ❛ s✐♠♣❧❡ ❣r❛♣❤ ♦❢ ♠❛①✐♠✉♠ ❞❡❣r❡❡ ∆( G ) ✳ ❚❤❡ ♠✐♥✐♠✉♠ ♥✉♠❜❡r ♦❢ ❝♦❧♦rs ♥❡❡❞❡❞ t♦ ❝♦❧♦r ❛❧❧ ❡❞❣❡s ♦❢ G ✐s ❡✐t❤❡r ∆( G ) ♦r ∆( G ) + 1 ✳ ❏❡s♣❡r ❲✳ ▼✐❦❦❡❧s❡♥ ✸✴✸✹

❆ ✜①❡❞ ♥✉♠❜❡r✱ ✱ ♦❢ ❝♦❧♦rs ✐s ❛✈❛✐❧❛❜❧❡✳ ❚❤❡ ❣♦❛❧ ✐s t♦ ❝♦❧♦r ❛s ♠❛♥② ❡❞❣❡s ❛s ♣♦ss✐❜❧❡✳ ❲❡ ❧❛❜❡❧ t❤❡ ❝♦❧♦rs ✳ ❋♦r ✱ t❤✐s ✐s t❤❡ ♠❛①✐♠✉♠ ♠❛t❝❤✐♥❣ ♣r♦❜❧❡♠✳ ❉✉❛❧ ❊❞❣❡ ❈♦❧♦r✐♥❣ ❚❤❡r❡ ✐s ❛ ❞✉❛❧ ✈❡rs✐♦♥ ❦♥♦✇♥ ❛s ❊❞❣❡✲ k ✲❈♦❧♦r✐♥❣✿ ❏❡s♣❡r ❲✳ ▼✐❦❦❡❧s❡♥ ✹✴✸✹

❲❡ ❧❛❜❡❧ t❤❡ ❝♦❧♦rs ✳ ❋♦r ✱ t❤✐s ✐s t❤❡ ♠❛①✐♠✉♠ ♠❛t❝❤✐♥❣ ♣r♦❜❧❡♠✳ ❉✉❛❧ ❊❞❣❡ ❈♦❧♦r✐♥❣ ❚❤❡r❡ ✐s ❛ ❞✉❛❧ ✈❡rs✐♦♥ ❦♥♦✇♥ ❛s ❊❞❣❡✲ k ✲❈♦❧♦r✐♥❣✿ ◮ ❆ ✜①❡❞ ♥✉♠❜❡r✱ k ✱ ♦❢ ❝♦❧♦rs ✐s ❛✈❛✐❧❛❜❧❡✳ ◮ ❚❤❡ ❣♦❛❧ ✐s t♦ ❝♦❧♦r ❛s ♠❛♥② ❡❞❣❡s ❛s ♣♦ss✐❜❧❡✳ ❏❡s♣❡r ❲✳ ▼✐❦❦❡❧s❡♥ ✹✴✸✹

❋♦r ✱ t❤✐s ✐s t❤❡ ♠❛①✐♠✉♠ ♠❛t❝❤✐♥❣ ♣r♦❜❧❡♠✳ ❉✉❛❧ ❊❞❣❡ ❈♦❧♦r✐♥❣ ❚❤❡r❡ ✐s ❛ ❞✉❛❧ ✈❡rs✐♦♥ ❦♥♦✇♥ ❛s ❊❞❣❡✲ k ✲❈♦❧♦r✐♥❣✿ ◮ ❆ ✜①❡❞ ♥✉♠❜❡r✱ k ✱ ♦❢ ❝♦❧♦rs ✐s ❛✈❛✐❧❛❜❧❡✳ ◮ ❚❤❡ ❣♦❛❧ ✐s t♦ ❝♦❧♦r ❛s ♠❛♥② ❡❞❣❡s ❛s ♣♦ss✐❜❧❡✳ ❲❡ ❧❛❜❡❧ t❤❡ k ❝♦❧♦rs 1 , 2 , . . . , k ✳ ❏❡s♣❡r ❲✳ ▼✐❦❦❡❧s❡♥ ✹✴✸✹

❉✉❛❧ ❊❞❣❡ ❈♦❧♦r✐♥❣ ❚❤❡r❡ ✐s ❛ ❞✉❛❧ ✈❡rs✐♦♥ ❦♥♦✇♥ ❛s ❊❞❣❡✲ k ✲❈♦❧♦r✐♥❣✿ ◮ ❆ ✜①❡❞ ♥✉♠❜❡r✱ k ✱ ♦❢ ❝♦❧♦rs ✐s ❛✈❛✐❧❛❜❧❡✳ ◮ ❚❤❡ ❣♦❛❧ ✐s t♦ ❝♦❧♦r ❛s ♠❛♥② ❡❞❣❡s ❛s ♣♦ss✐❜❧❡✳ ❲❡ ❧❛❜❡❧ t❤❡ k ❝♦❧♦rs 1 , 2 , . . . , k ✳ ❋♦r k = 1 ✱ t❤✐s ✐s t❤❡ ♠❛①✐♠✉♠ ♠❛t❝❤✐♥❣ ♣r♦❜❧❡♠✳ ❏❡s♣❡r ❲✳ ▼✐❦❦❡❧s❡♥ ✹✴✸✹

❖♥❧✐♥❡ ❊❞❣❡ ❈♦❧♦r✐♥❣ ❖♥❧✐♥❡ ❊❞❣❡✲ k ✲❈♦❧♦r✐♥❣ ◮ ❊❞❣❡s ❛rr✐✈❡ ♦♥❡ ❜② ♦♥❡✳ ◮ ▼✉st ✐♠♠❡❞✐❛t❡❧② ❝♦❧♦r ❛ ♥❡✇❧② ❛rr✐✈❡❞ ❡❞❣❡ ✇✐t❤ ♦♥❡ ♦❢ t❤❡ k ❝♦❧♦rs ♦r r❡❥❡❝t t❤❡ ❡❞❣❡✳ ◮ ❚❤❡ ❞❡❝✐s✐♦♥ ✐s ✐rr❡✈♦❝❛❜❧❡ ✳ ❏❡s♣❡r ❲✳ ▼✐❦❦❡❧s❡♥ ✺✴✸✹

❊①❛♠♣❧❡ ❢♦r k = 2 2 1 2 1 ❏❡s♣❡r ❲✳ ▼✐❦❦❡❧s❡♥ ✻✴✸✹

❊①❛♠♣❧❡ ❢♦r k = 2 1 2 1 2 1 ❏❡s♣❡r ❲✳ ▼✐❦❦❡❧s❡♥ ✻✴✸✹

❊①❛♠♣❧❡ ❢♦r k = 2 1 2 2 1 ❏❡s♣❡r ❲✳ ▼✐❦❦❡❧s❡♥ ✻✴✸✹

❊①❛♠♣❧❡ ❢♦r k = 2 1 2 1 2 1 ❏❡s♣❡r ❲✳ ▼✐❦❦❡❧s❡♥ ✻✴✸✹

❊①❛♠♣❧❡ ❢♦r k = 2 1 1 2 1 ❏❡s♣❡r ❲✳ ▼✐❦❦❡❧s❡♥ ✻✴✸✹

❊①❛♠♣❧❡ ❢♦r k = 2 1 2 1 2 1 ❏❡s♣❡r ❲✳ ▼✐❦❦❡❧s❡♥ ✻✴✸✹

❊①❛♠♣❧❡ ❢♦r k = 2 1 2 1 2 1 ❏❡s♣❡r ❲✳ ▼✐❦❦❡❧s❡♥ ✻✴✸✹

❊①❛♠♣❧❡ ❢♦r k = 2 1 2 1 2 1 ❏❡s♣❡r ❲✳ ▼✐❦❦❡❧s❡♥ ✻✴✸✹

❖♥❧✐♥❡ ❊❞❣❡ ❈♦❧♦r✐♥❣ ❈♦♠♣❡t✐t✐✈❡ ❛♥❛❧②s✐s ❬❙❧❡❛t♦r✱ ❚❛r❥❛♥ ✬✽✺❪✱ ❬❑❛r❧✐♥ ❡t ❛❧✳ ✬✽✽❪ ❆♥ ❛❧❣♦r✐t❤♠ ❆ ✐s c ✲❝♦♠♣❡t✐t✐✈❡ ✐❢ ❆ ( σ ) ≥ c · ❖P❚ ( σ ) − b ❢♦r ❛❧❧ s❡q✉❡♥❝❡ ♦❢ ❡❞❣❡s σ ✳ ❋♦r ❛ r❛♥❞♦♠✐③❡❞ ❛❧❣♦r✐t❤♠✱ r❡♣❧❛❝❡ ❆ ( σ ) ✇✐t❤ E [ ❆ ( σ )] ✳ ❚❤❡ ❝♦♠♣❡t✐t✐✈❡ r❛t✐♦ C = sup { c : ❆ ✐s c ✲❝♦♠♣❡t✐t✐✈❡ } ✳ ❏❡s♣❡r ❲✳ ▼✐❦❦❡❧s❡♥ ✼✴✸✹

❖♥❧✐♥❡ ❊❞❣❡ ❈♦❧♦r✐♥❣ ❈♦♠♣❡t✐t✐✈❡ ❛♥❛❧②s✐s ❬❙❧❡❛t♦r✱ ❚❛r❥❛♥ ✬✽✺❪✱ ❬❑❛r❧✐♥ ❡t ❛❧✳ ✬✽✽❪ ❆♥ ❛❧❣♦r✐t❤♠ ❆ ✐s c ✲❝♦♠♣❡t✐t✐✈❡ ✐❢ ❆ ( σ ) ≥ c · ❖P❚ ( σ ) − b ❢♦r ❛❧❧ s❡q✉❡♥❝❡ ♦❢ ❡❞❣❡s σ ✳ ❋♦r ❛ r❛♥❞♦♠✐③❡❞ ❛❧❣♦r✐t❤♠✱ r❡♣❧❛❝❡ ❆ ( σ ) ✇✐t❤ E [ ❆ ( σ )] ✳ ❚❤❡ ❝♦♠♣❡t✐t✐✈❡ r❛t✐♦ C = sup { c : ❆ ✐s c ✲❝♦♠♣❡t✐t✐✈❡ } ✳ ❏❡s♣❡r ❲✳ ▼✐❦❦❡❧s❡♥ ✼✴✸✹

P♦s✐t✐✈❡ r❡s✉❧ts✿ ❚❤❡ ❝♦♠♣❡t✐t✐✈❡ r❛t✐♦ ♦❢ ❛ ❢❛✐r ❛❧❣♦r✐t❤♠ ✐s ❛t ❧❡❛st ❆♥ ❛❧❣♦r✐t❤♠ ✐s ❢❛✐r ✐❢ ✐t ♥❡✈❡r r❡❥❡❝ts ❛♥ ❡❞❣❡ ✉♥❧❡ss ❢♦r❝❡❞ t♦ ❞♦ s♦✳ Pr❡✈✐♦✉s r❡s✉❧ts ✭❋❛✈r❤♦❧❞t✱ ◆✐❡❧s❡♥ ✬✵✸✮ ◮ ◆❡❣❛t✐✈❡ r❡s✉❧ts✿ ◆♦ ❞❡t❡r♠✐♥✐st✐❝ ❛❧❣♦r✐t❤♠ ❤❛s ❛ ❝♦♠♣❡t✐t✐✈❡ r❛t✐♦ ❜❡tt❡r t❤❛♥ 1 2 ✳ ◆♦ r❛♥❞♦♠✐③❡❞ ❛❧❣♦r✐t❤♠ ❤❛s ❛ ❝♦♠♣❡t✐t✐✈❡ r❛t✐♦ ❜❡tt❡r t❤❛♥ 4 7 ✳ ❏❡s♣❡r ❲✳ ▼✐❦❦❡❧s❡♥ ✽✴✸✹

Pr❡✈✐♦✉s r❡s✉❧ts ✭❋❛✈r❤♦❧❞t✱ ◆✐❡❧s❡♥ ✬✵✸✮ ◮ ◆❡❣❛t✐✈❡ r❡s✉❧ts✿ ◆♦ ❞❡t❡r♠✐♥✐st✐❝ ❛❧❣♦r✐t❤♠ ❤❛s ❛ ❝♦♠♣❡t✐t✐✈❡ r❛t✐♦ ❜❡tt❡r t❤❛♥ 1 2 ✳ ◆♦ r❛♥❞♦♠✐③❡❞ ❛❧❣♦r✐t❤♠ ❤❛s ❛ ❝♦♠♣❡t✐t✐✈❡ r❛t✐♦ ❜❡tt❡r t❤❛♥ 4 7 ✳ ◮ P♦s✐t✐✈❡ r❡s✉❧ts✿ √ ❚❤❡ ❝♦♠♣❡t✐t✐✈❡ r❛t✐♦ ♦❢ ❛ ❢❛✐r ❛❧❣♦r✐t❤♠ ✐s ❛t ❧❡❛st 2 3 − 3 ≈ 0 . 46 ❆♥ ❛❧❣♦r✐t❤♠ ✐s ❢❛✐r ✐❢ ✐t ♥❡✈❡r r❡❥❡❝ts ❛♥ ❡❞❣❡ ✉♥❧❡ss ❢♦r❝❡❞ t♦ ❞♦ s♦✳ ❏❡s♣❡r ❲✳ ▼✐❦❦❡❧s❡♥ ✽✴✸✹