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SLIDE 1

❉❡❝✐❞✐♥❣ t❤❡ ❖♥✲❧✐♥❡ ❈❤r♦♠❛t✐❝ ◆✉♠❜❡r ♦❢ ❛ ●r❛♣❤ ✇✐t❤ Pr❡✲❈♦❧♦r✐♥❣ ✐s P❙P❆❈❊✲❈♦♠♣❧❡t❡

❈❤r✐st✐❛♥ ❑✉❞❛❤❧

❯♥✐✈❡rs✐t② ♦❢ ❙♦✉t❤❡r♥ ❉❡♥♠❛r❦ ❦✉❞❛❤❧❅✐♠❛❞❛✳s❞✉✳❞❦

❏✉❧② ✼✱ ✷✵✶✹

❈❤r✐st✐❛♥ ❑✉❞❛❤❧ ✭■▼❆❉❆✮ ❖♥✲❧✐♥❡ ❈❤r♦♠❛t✐❝ ◆✉♠❜❡r ❏✉❧② ✼✱ ✷✵✶✹ ✶ ✴ ✷✵

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SLIDE 2

❖♥✲❧✐♥❡ ❣r❛♣❤ ❝♦❧♦r✐♥❣

❚❤❡ ✈❡rt✐❝❡s ♦❢ ❛ ❣r❛♣❤ ❛r❡ r❡✈❡❛❧❡❞ ♦♥❡ ❜② ♦♥❡✳ ❲❤❡♥ ✈❡rt❡①✱ ✱ ✐s r❡✈❡❛❧❡❞✱ ✐t ✐s r❡✈❡❛❧❡❞ ✇❤✐❝❤ ✈❡rt✐❝❡s ✐t ✐s ❛❞❥❛❝❡♥t t♦ ✭❛♠♦♥❣ t❤♦s❡ t❤❛t ✇❡r❡ ♣r❡✈✐♦✉s❧② r❡✈❡❛❧❡❞✮✳ ❚❤❡ ❛❧❣♦r✐t❤♠ ❛ss✐❣♥s ❛ ❝♦❧♦r t♦ ❞✐✛❡r❡♥t ❢r♦♠ t❤❡ ❝♦❧♦rs ✐t ✐s ❛❧r❡❛❞② ❛❞❥❛❝❡♥t t♦✳ ❚❤❡ ❝♦❧♦r ❝❛♥♥♦t ❜❡ ❝❤❛♥❣❡❞ ❧❛t❡r✳ ❚❤❡ ❣♦❛❧ ✐s t♦ ✉s❡ ❛s ❢❡✇ ❝♦❧♦rs ❛s ♣♦ss✐❜❧❡✳

❈❤r✐st✐❛♥ ❑✉❞❛❤❧ ✭■▼❆❉❆✮ ❖♥✲❧✐♥❡ ❈❤r♦♠❛t✐❝ ◆✉♠❜❡r ❏✉❧② ✼✱ ✷✵✶✹ ✷ ✴ ✷✵

slide-3
SLIDE 3

❖♥✲❧✐♥❡ ❣r❛♣❤ ❝♦❧♦r✐♥❣

❚❤❡ ✈❡rt✐❝❡s ♦❢ ❛ ❣r❛♣❤ ❛r❡ r❡✈❡❛❧❡❞ ♦♥❡ ❜② ♦♥❡✳ ❲❤❡♥ ✈❡rt❡①✱ v✱ ✐s r❡✈❡❛❧❡❞✱ ✐t ✐s r❡✈❡❛❧❡❞ ✇❤✐❝❤ ✈❡rt✐❝❡s ✐t ✐s ❛❞❥❛❝❡♥t t♦ ✭❛♠♦♥❣ t❤♦s❡ t❤❛t ✇❡r❡ ♣r❡✈✐♦✉s❧② r❡✈❡❛❧❡❞✮✳ ❚❤❡ ❛❧❣♦r✐t❤♠ ❛ss✐❣♥s ❛ ❝♦❧♦r t♦ ❞✐✛❡r❡♥t ❢r♦♠ t❤❡ ❝♦❧♦rs ✐t ✐s ❛❧r❡❛❞② ❛❞❥❛❝❡♥t t♦✳ ❚❤❡ ❝♦❧♦r ❝❛♥♥♦t ❜❡ ❝❤❛♥❣❡❞ ❧❛t❡r✳ ❚❤❡ ❣♦❛❧ ✐s t♦ ✉s❡ ❛s ❢❡✇ ❝♦❧♦rs ❛s ♣♦ss✐❜❧❡✳

❈❤r✐st✐❛♥ ❑✉❞❛❤❧ ✭■▼❆❉❆✮ ❖♥✲❧✐♥❡ ❈❤r♦♠❛t✐❝ ◆✉♠❜❡r ❏✉❧② ✼✱ ✷✵✶✹ ✷ ✴ ✷✵

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SLIDE 4

❖♥✲❧✐♥❡ ❣r❛♣❤ ❝♦❧♦r✐♥❣

❚❤❡ ✈❡rt✐❝❡s ♦❢ ❛ ❣r❛♣❤ ❛r❡ r❡✈❡❛❧❡❞ ♦♥❡ ❜② ♦♥❡✳ ❲❤❡♥ ✈❡rt❡①✱ v✱ ✐s r❡✈❡❛❧❡❞✱ ✐t ✐s r❡✈❡❛❧❡❞ ✇❤✐❝❤ ✈❡rt✐❝❡s ✐t ✐s ❛❞❥❛❝❡♥t t♦ ✭❛♠♦♥❣ t❤♦s❡ t❤❛t ✇❡r❡ ♣r❡✈✐♦✉s❧② r❡✈❡❛❧❡❞✮✳ ❚❤❡ ❛❧❣♦r✐t❤♠ ❛ss✐❣♥s ❛ ❝♦❧♦r t♦ v ❞✐✛❡r❡♥t ❢r♦♠ t❤❡ ❝♦❧♦rs ✐t ✐s ❛❧r❡❛❞② ❛❞❥❛❝❡♥t t♦✳ ❚❤❡ ❝♦❧♦r ❝❛♥♥♦t ❜❡ ❝❤❛♥❣❡❞ ❧❛t❡r✳ ❚❤❡ ❣♦❛❧ ✐s t♦ ✉s❡ ❛s ❢❡✇ ❝♦❧♦rs ❛s ♣♦ss✐❜❧❡✳

❈❤r✐st✐❛♥ ❑✉❞❛❤❧ ✭■▼❆❉❆✮ ❖♥✲❧✐♥❡ ❈❤r♦♠❛t✐❝ ◆✉♠❜❡r ❏✉❧② ✼✱ ✷✵✶✹ ✷ ✴ ✷✵

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SLIDE 5

❖♥✲❧✐♥❡ ❣r❛♣❤ ❝♦❧♦r✐♥❣

❚❤❡ ✈❡rt✐❝❡s ♦❢ ❛ ❣r❛♣❤ ❛r❡ r❡✈❡❛❧❡❞ ♦♥❡ ❜② ♦♥❡✳ ❲❤❡♥ ✈❡rt❡①✱ v✱ ✐s r❡✈❡❛❧❡❞✱ ✐t ✐s r❡✈❡❛❧❡❞ ✇❤✐❝❤ ✈❡rt✐❝❡s ✐t ✐s ❛❞❥❛❝❡♥t t♦ ✭❛♠♦♥❣ t❤♦s❡ t❤❛t ✇❡r❡ ♣r❡✈✐♦✉s❧② r❡✈❡❛❧❡❞✮✳ ❚❤❡ ❛❧❣♦r✐t❤♠ ❛ss✐❣♥s ❛ ❝♦❧♦r t♦ v ❞✐✛❡r❡♥t ❢r♦♠ t❤❡ ❝♦❧♦rs ✐t ✐s ❛❧r❡❛❞② ❛❞❥❛❝❡♥t t♦✳ ❚❤❡ ❝♦❧♦r ❝❛♥♥♦t ❜❡ ❝❤❛♥❣❡❞ ❧❛t❡r✳ ❚❤❡ ❣♦❛❧ ✐s t♦ ✉s❡ ❛s ❢❡✇ ❝♦❧♦rs ❛s ♣♦ss✐❜❧❡✳

❈❤r✐st✐❛♥ ❑✉❞❛❤❧ ✭■▼❆❉❆✮ ❖♥✲❧✐♥❡ ❈❤r♦♠❛t✐❝ ◆✉♠❜❡r ❏✉❧② ✼✱ ✷✵✶✹ ✷ ✴ ✷✵

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SLIDE 6

❆♥ ❡①❛♠♣❧❡

❈❤r✐st✐❛♥ ❑✉❞❛❤❧ ✭■▼❆❉❆✮ ❖♥✲❧✐♥❡ ❈❤r♦♠❛t✐❝ ◆✉♠❜❡r ❏✉❧② ✼✱ ✷✵✶✹ ✸ ✴ ✷✵

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SLIDE 7

❆♥ ❡①❛♠♣❧❡

❈❤r✐st✐❛♥ ❑✉❞❛❤❧ ✭■▼❆❉❆✮ ❖♥✲❧✐♥❡ ❈❤r♦♠❛t✐❝ ◆✉♠❜❡r ❏✉❧② ✼✱ ✷✵✶✹ ✸ ✴ ✷✵

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SLIDE 8

❆♥ ❡①❛♠♣❧❡

❈❤r✐st✐❛♥ ❑✉❞❛❤❧ ✭■▼❆❉❆✮ ❖♥✲❧✐♥❡ ❈❤r♦♠❛t✐❝ ◆✉♠❜❡r ❏✉❧② ✼✱ ✷✵✶✹ ✸ ✴ ✷✵

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SLIDE 9

❆♥ ❡①❛♠♣❧❡

❈❤r✐st✐❛♥ ❑✉❞❛❤❧ ✭■▼❆❉❆✮ ❖♥✲❧✐♥❡ ❈❤r♦♠❛t✐❝ ◆✉♠❜❡r ❏✉❧② ✼✱ ✷✵✶✹ ✸ ✴ ✷✵

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SLIDE 10

❆♥ ❡①❛♠♣❧❡

✶ ✶

❈❤r✐st✐❛♥ ❑✉❞❛❤❧ ✭■▼❆❉❆✮ ❖♥✲❧✐♥❡ ❈❤r♦♠❛t✐❝ ◆✉♠❜❡r ❏✉❧② ✼✱ ✷✵✶✹ ✸ ✴ ✷✵

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SLIDE 11

❆♥ ❡①❛♠♣❧❡

✶ ✶

❈❤r✐st✐❛♥ ❑✉❞❛❤❧ ✭■▼❆❉❆✮ ❖♥✲❧✐♥❡ ❈❤r♦♠❛t✐❝ ◆✉♠❜❡r ❏✉❧② ✼✱ ✷✵✶✹ ✸ ✴ ✷✵

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SLIDE 12

❆♥ ❡①❛♠♣❧❡

✶ ✷ ✶

❈❤r✐st✐❛♥ ❑✉❞❛❤❧ ✭■▼❆❉❆✮ ❖♥✲❧✐♥❡ ❈❤r♦♠❛t✐❝ ◆✉♠❜❡r ❏✉❧② ✼✱ ✷✵✶✹ ✸ ✴ ✷✵

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SLIDE 13

❆♥ ❡①❛♠♣❧❡

✶ ✷ ✶

❈❤r✐st✐❛♥ ❑✉❞❛❤❧ ✭■▼❆❉❆✮ ❖♥✲❧✐♥❡ ❈❤r♦♠❛t✐❝ ◆✉♠❜❡r ❏✉❧② ✼✱ ✷✵✶✹ ✸ ✴ ✷✵

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SLIDE 14

❆♥ ❡①❛♠♣❧❡

✶ ✷ ✸ ✶

❈❤r✐st✐❛♥ ❑✉❞❛❤❧ ✭■▼❆❉❆✮ ❖♥✲❧✐♥❡ ❈❤r♦♠❛t✐❝ ◆✉♠❜❡r ❏✉❧② ✼✱ ✷✵✶✹ ✸ ✴ ✷✵

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SLIDE 15

❍♦✇ t♦ ♠❡❛s✉r❡ t❤❡ q✉❛❧✐t② ♦❢ ❛♥ ❛❧❣♦r✐t❤♠❄

❈♦♠♣❡t✐t✐✈❡ ❆♥❛❧②s✐s ❬❙❧❡❛t♦r✱ ❚❛r❥❛♥ ✽✺❪

❆♥ ❛❧❣♦r✐t❤♠ A ✐s c✲❝♦♠♣❡t✐t✐✈❡ ✐❢ t❤❡r❡ ❡①✐sts ❛ b s✉❝❤ t❤❛t ❢♦r ❛♥② ❣r❛♣❤✱ G✱ t❤❡ ❢♦❧❧♦✇✐♥❣ ❤♦❧❞s✿ A(G) ≤ c · χ(G) + b ❍❡r❡✱ χ(G) ✐s t❤❡ ❝❤r♦♠❛t✐❝ ♥✉♠❜❡r ♦❢ G✳ ❚❤❛t ✐s t❤❡ ♠✐♥✐♠✉♠ ♥✉♠❜❡r ♦❢ ❝♦❧♦rs ♥❡❡❞❡❞ t♦ ❝♦❧♦r G ✭♦✤✐♥❡✮✳

❈❤r✐st✐❛♥ ❑✉❞❛❤❧ ✭■▼❆❉❆✮ ❖♥✲❧✐♥❡ ❈❤r♦♠❛t✐❝ ◆✉♠❜❡r ❏✉❧② ✼✱ ✷✵✶✹ ✹ ✴ ✷✵

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SLIDE 16

❍♦✇ t♦ ♠❡❛s✉r❡ t❤❡ q✉❛❧✐t② ♦❢ ❛♥ ❛❧❣♦r✐t❤♠❄

❚❤❡♦r❡♠ ❬●②❛r❢❛s✱ ❑✐r❛❧②✱ ▲❡❤❡❧ ✶✾✾✵❪

❋♦r ❛♥② k ∈ N✱ t❤❡r❡ ❡①✐sts ❛ tr❡❡ Tk s✉❝❤ t❤❛t ❛♥② ♦♥✲❧✐♥❡ ❛❧❣♦r✐t❤♠ ❝❛♥ ❜❡ ❢♦r❝❡❞ t♦ ✉s❡ k ❝♦❧♦rs ✇❤❡♥ ♦♥✲❧✐♥❡ ❝♦❧♦r✐♥❣ Tk✳

❋✐❣✉r❡✿ T✸ ❛♥❞ T✹✳

❚❤✐s ♠❡❛♥s t❤❛t ♥♦ c✲❝♦♠♣❡t✐t✐✈❡ ❛❧❣♦r✐t❤♠s ❡①✐st ❢♦r ❛♥② ❝♦♥st❛♥t c ✲ ❡✈❡♥ ❢♦r t❤❡ ❝❧❛ss ♦❢ tr❡❡s✳

❈❤r✐st✐❛♥ ❑✉❞❛❤❧ ✭■▼❆❉❆✮ ❖♥✲❧✐♥❡ ❈❤r♦♠❛t✐❝ ◆✉♠❜❡r ❏✉❧② ✼✱ ✷✵✶✹ ✺ ✴ ✷✵

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SLIDE 17

❍♦✇ t♦ ♠❡❛s✉r❡ t❤❡ q✉❛❧✐t② ♦❢ ❛♥ ❛❧❣♦r✐t❤♠❄

❖♥✲❧✐♥❡ ❈♦♠♣❡t✐t✐✈❡ ❆♥❛❧②s✐s ❬●②❛r❢❛s✱ ❑✐r❛❧②✱ ▲❡❤❡❧❪

❆♥ ❛❧❣♦r✐t❤♠ A ✐s ♦♥✲❧✐♥❡ c✲❝♦♠♣❡t✐t✐✈❡ ✐❢ t❤❡r❡ ❡①✐sts ❛♥ b s✉❝❤ t❤❛t ❢♦r ❛♥② ❣r❛♣❤✱ G✱ t❤❡ ❢♦❧❧♦✇✐♥❣ ❤♦❧❞s✿ A(G) ≤ c · χO(G) + b ❍❡r❡✱ χO(G) ✐s t❤❡ ♦♥✲❧✐♥❡ ❝❤r♦♠❛t✐❝ ♥✉♠❜❡r ♦❢ G✳ ❚❤✐s ✐s t❤❡ s♠❛❧❧❡st ♥✉♠❜❡r s✉❝❤ t❤❛t t❤❡r❡ ❡①✐sts ❛♥ ❛❧❣♦r✐t❤♠ t❤❛t ❝❛♥ ❝♦❧♦r G ✉s✐♥❣ ❛t ♠♦st χO(G) ❝♦❧♦rs ❢♦r ❡✈❡r② ♦r❞❡r✐♥❣ ♦❢ t❤❡ ✈❡rt✐❝❡s✳

❈❤r✐st✐❛♥ ❑✉❞❛❤❧ ✭■▼❆❉❆✮ ❖♥✲❧✐♥❡ ❈❤r♦♠❛t✐❝ ◆✉♠❜❡r ❏✉❧② ✼✱ ✷✵✶✹ ✻ ✴ ✷✵

slide-18
SLIDE 18

❖♥✲❧✐♥❡ ❈❤r♦♠❛t✐❝ ◆✉♠❜❡r

✶ ❋✐❣✉r❡✿ ❖♥✲❧✐♥❡ ❝♦❧♦r✐♥❣ G✳ ❚♦ t❤❡ r✐❣❤t✱ G ✐s s❤♦✇♥✳

❈❤r✐st✐❛♥ ❑✉❞❛❤❧ ✭■▼❆❉❆✮ ❖♥✲❧✐♥❡ ❈❤r♦♠❛t✐❝ ◆✉♠❜❡r ❏✉❧② ✼✱ ✷✵✶✹ ✼ ✴ ✷✵

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SLIDE 19

❖♥✲❧✐♥❡ ❈❤r♦♠❛t✐❝ ◆✉♠❜❡r

✶ ✷ ❋✐❣✉r❡✿ ❖♥✲❧✐♥❡ ❝♦❧♦r✐♥❣ G✳ ❚♦ t❤❡ r✐❣❤t✱ G ✐s s❤♦✇♥✳

❈❤r✐st✐❛♥ ❑✉❞❛❤❧ ✭■▼❆❉❆✮ ❖♥✲❧✐♥❡ ❈❤r♦♠❛t✐❝ ◆✉♠❜❡r ❏✉❧② ✼✱ ✷✵✶✹ ✼ ✴ ✷✵

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SLIDE 20

❖♥✲❧✐♥❡ ❈❤r♦♠❛t✐❝ ◆✉♠❜❡r

✶ ✷ ✶ ❋✐❣✉r❡✿ ❖♥✲❧✐♥❡ ❝♦❧♦r✐♥❣ G✳ ❚♦ t❤❡ r✐❣❤t✱ G ✐s s❤♦✇♥✳

❈❤r✐st✐❛♥ ❑✉❞❛❤❧ ✭■▼❆❉❆✮ ❖♥✲❧✐♥❡ ❈❤r♦♠❛t✐❝ ◆✉♠❜❡r ❏✉❧② ✼✱ ✷✵✶✹ ✼ ✴ ✷✵

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SLIDE 21

❖♥✲❧✐♥❡ ❈❤r♦♠❛t✐❝ ◆✉♠❜❡r

✶ ✷ ✶ ✷ ❋✐❣✉r❡✿ ❖♥✲❧✐♥❡ ❝♦❧♦r✐♥❣ G✳ ❚♦ t❤❡ r✐❣❤t✱ G ✐s s❤♦✇♥✳

❈❤r✐st✐❛♥ ❑✉❞❛❤❧ ✭■▼❆❉❆✮ ❖♥✲❧✐♥❡ ❈❤r♦♠❛t✐❝ ◆✉♠❜❡r ❏✉❧② ✼✱ ✷✵✶✹ ✼ ✴ ✷✵

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SLIDE 22

❖♥✲❧✐♥❡ ❈❤r♦♠❛t✐❝ ◆✉♠❜❡r

✶ ✷ ✸ ✶ ✷ ❋✐❣✉r❡✿ ❖♥✲❧✐♥❡ ❝♦❧♦r✐♥❣ G✳ ❚♦ t❤❡ r✐❣❤t✱ G ✐s s❤♦✇♥✳

❈❤r✐st✐❛♥ ❑✉❞❛❤❧ ✭■▼❆❉❆✮ ❖♥✲❧✐♥❡ ❈❤r♦♠❛t✐❝ ◆✉♠❜❡r ❏✉❧② ✼✱ ✷✵✶✹ ✼ ✴ ✷✵

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SLIDE 23

❖♥✲❧✐♥❡ ❈❤r♦♠❛t✐❝ ◆✉♠❜❡r

✶ ✷ ✸ ✹ ✶ ✷ ❋✐❣✉r❡✿ ❖♥✲❧✐♥❡ ❝♦❧♦r✐♥❣ G✳ ❚♦ t❤❡ r✐❣❤t✱ G ✐s s❤♦✇♥✳

❈❤r✐st✐❛♥ ❑✉❞❛❤❧ ✭■▼❆❉❆✮ ❖♥✲❧✐♥❡ ❈❤r♦♠❛t✐❝ ◆✉♠❜❡r ❏✉❧② ✼✱ ✷✵✶✹ ✼ ✴ ✷✵

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SLIDE 24

❖♥✲❧✐♥❡ ❈❤r♦♠❛t✐❝ ◆✉♠❜❡r

✶ ✷ ✸ ✹ ✶ ✷ ❋✐❣✉r❡✿ ❖♥✲❧✐♥❡ ❝♦❧♦r✐♥❣ G✳ ❚♦ t❤❡ r✐❣❤t✱ G ✐s s❤♦✇♥✳

■t t✉r♥s ♦✉t t❤❛t χO(G) = ✸ ✭t❤❡ ❜♦tt♦♠ r✐❣❤t ✈❡rt❡① s❤♦✉❧❞ ❤❛✈❡ ❜❡❡♥ ❣✐✈❡♥ t❤❡ ❝♦❧♦r ✸ ✐♥st❡❛❞✮✳

❈❤r✐st✐❛♥ ❑✉❞❛❤❧ ✭■▼❆❉❆✮ ❖♥✲❧✐♥❡ ❈❤r♦♠❛t✐❝ ◆✉♠❜❡r ❏✉❧② ✼✱ ✷✵✶✹ ✼ ✴ ✷✵

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SLIDE 25

❖♥✲❧✐♥❡ ❈❤r♦♠❛t✐❝ ◆✉♠❜❡r

✶ ✷ ✶ ✸ ❋✐❣✉r❡✿ ❖♥✲❧✐♥❡ ❝♦❧♦r✐♥❣ G✳ ❚♦ t❤❡ r✐❣❤t✱ G ✐s s❤♦✇♥✳

❈❤r✐st✐❛♥ ❑✉❞❛❤❧ ✭■▼❆❉❆✮ ❖♥✲❧✐♥❡ ❈❤r♦♠❛t✐❝ ◆✉♠❜❡r ❏✉❧② ✼✱ ✷✵✶✹ ✼ ✴ ✷✵

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SLIDE 26

❖♥✲❧✐♥❡ ❈❤r♦♠❛t✐❝ ◆✉♠❜❡r

✶ ✷ ✷ ✶ ✸ ❋✐❣✉r❡✿ ❖♥✲❧✐♥❡ ❝♦❧♦r✐♥❣ G✳ ❚♦ t❤❡ r✐❣❤t✱ G ✐s s❤♦✇♥✳

❈❤r✐st✐❛♥ ❑✉❞❛❤❧ ✭■▼❆❉❆✮ ❖♥✲❧✐♥❡ ❈❤r♦♠❛t✐❝ ◆✉♠❜❡r ❏✉❧② ✼✱ ✷✵✶✹ ✼ ✴ ✷✵

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SLIDE 27

❖♥✲❧✐♥❡ ❈❤r♦♠❛t✐❝ ◆✉♠❜❡r

✶ ✷ ✷ ✸ ✶ ✸ ❋✐❣✉r❡✿ ❖♥✲❧✐♥❡ ❝♦❧♦r✐♥❣ G✳ ❚♦ t❤❡ r✐❣❤t✱ G ✐s s❤♦✇♥✳

❈❤r✐st✐❛♥ ❑✉❞❛❤❧ ✭■▼❆❉❆✮ ❖♥✲❧✐♥❡ ❈❤r♦♠❛t✐❝ ◆✉♠❜❡r ❏✉❧② ✼✱ ✷✵✶✹ ✼ ✴ ✷✵

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SLIDE 28

❚❤❡ ❖♥✲❧✐♥❡ ❈❤r♦♠❛t✐❝ ◆✉♠❜❡r

❲❡ ❦♥♦✇ t❤❛t ✐t ✐s ◆P✲❝♦♠♣❧❡t❡ ❣✐✈❡♥ ❛ k ∈ N ❛♥❞ ❛ ❣r❛♣❤ G t♦ ❞❡❝✐❞❡ ✐❢ χ(G) ≤ k✳

▼❛✐♥ Pr♦❜❧❡♠

❍♦✇ ❤❛r❞ ✐s ✐t ❣✐✈❡♥ ❛ k ∈ N ❛♥❞ ❛ ❣r❛♣❤ G t♦ ❞❡❝✐❞❡ ✐❢ χO(G) ≤ k❄

❈❤r✐st✐❛♥ ❑✉❞❛❤❧ ✭■▼❆❉❆✮ ❖♥✲❧✐♥❡ ❈❤r♦♠❛t✐❝ ◆✉♠❜❡r ❏✉❧② ✼✱ ✷✵✶✹ ✽ ✴ ✷✵

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SLIDE 29

❚❤❡ ❖♥✲❧✐♥❡ ❈❤r♦♠❛t✐❝ ◆✉♠❜❡r

❖♥✲❧✐♥❡ ●r❛♣❤ ❝♦❧♦r✐♥❣ ❝❛♥ ❜❡ s❡❡♥ ❛s ❛ ❣❛♠❡ ❜❡t✇❡❡♥ t❤❡ ❉r❛✇❡r ❛♥❞ t❤❡ P❛✐♥t❡r✳ (G, k) ✐s t❤❡ ❣❛♠❡ ✐♥st❛♥❝❡ t❤❡② ❛❣r❡❡ ♦♥ ❜❡❢♦r❡ t❤❡ ❣❛♠❡✳ ❚❤❡ ❉r❛✇❡r ♣r❡s❡♥ts ❛ ✈❡rt❡① ♦♥ ❤✐s t✉r♥✳ ❚❤❡ P❛✐♥t❡r ❝♦❧♦rs t❤❛t ✈❡rt❡① ♦♥ ❤✐s t✉r♥✳ ❆t ❡❛❝❤ t✐♠❡✱ t❤❡ ♣r❡s❡♥t❡❞ ❣r❛♣❤ ✐s ✐s♦♠♦r♣❤✐❝ t♦ ❛♥ ✐♥❞✉❝❡❞ s✉❜❣r❛♣❤ ♦❢ ✳ ■❢ t❤❡ ♣❛✐♥t❡r ✐s ❛❜❧❡ t♦ ❝♦❧♦r t❤❡ ❡♥t✐r❡ ❣r❛♣❤ ✇✐t❤♦✉t ✉s✐♥❣ ♠♦r❡ t❤❛♥ ❝♦❧♦rs✱ ❤❡ ✇✐♥s✳ ❖t❤❡r✇✐s❡✱ ❤❡ ❧♦s❡s✳ ❚❤❡ ♣❛✐♥t❡r ❤❛s ❛ ✇✐♥♥✐♥❣ str❛t❡❣② ✐❢ ❛♥❞ ♦♥❧② ✐❢ ✳

❈❤r✐st✐❛♥ ❑✉❞❛❤❧ ✭■▼❆❉❆✮ ❖♥✲❧✐♥❡ ❈❤r♦♠❛t✐❝ ◆✉♠❜❡r ❏✉❧② ✼✱ ✷✵✶✹ ✾ ✴ ✷✵

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SLIDE 30

❚❤❡ ❖♥✲❧✐♥❡ ❈❤r♦♠❛t✐❝ ◆✉♠❜❡r

❖♥✲❧✐♥❡ ●r❛♣❤ ❝♦❧♦r✐♥❣ ❝❛♥ ❜❡ s❡❡♥ ❛s ❛ ❣❛♠❡ ❜❡t✇❡❡♥ t❤❡ ❉r❛✇❡r ❛♥❞ t❤❡ P❛✐♥t❡r✳ (G, k) ✐s t❤❡ ❣❛♠❡ ✐♥st❛♥❝❡ t❤❡② ❛❣r❡❡ ♦♥ ❜❡❢♦r❡ t❤❡ ❣❛♠❡✳ ❚❤❡ ❉r❛✇❡r ♣r❡s❡♥ts ❛ ✈❡rt❡① ♦♥ ❤✐s t✉r♥✳ ❚❤❡ P❛✐♥t❡r ❝♦❧♦rs t❤❛t ✈❡rt❡① ♦♥ ❤✐s t✉r♥✳ ❆t ❡❛❝❤ t✐♠❡✱ t❤❡ ♣r❡s❡♥t❡❞ ❣r❛♣❤ ✐s ✐s♦♠♦r♣❤✐❝ t♦ ❛♥ ✐♥❞✉❝❡❞ s✉❜❣r❛♣❤ ♦❢ ✳ ■❢ t❤❡ ♣❛✐♥t❡r ✐s ❛❜❧❡ t♦ ❝♦❧♦r t❤❡ ❡♥t✐r❡ ❣r❛♣❤ ✇✐t❤♦✉t ✉s✐♥❣ ♠♦r❡ t❤❛♥ ❝♦❧♦rs✱ ❤❡ ✇✐♥s✳ ❖t❤❡r✇✐s❡✱ ❤❡ ❧♦s❡s✳ ❚❤❡ ♣❛✐♥t❡r ❤❛s ❛ ✇✐♥♥✐♥❣ str❛t❡❣② ✐❢ ❛♥❞ ♦♥❧② ✐❢ ✳

❈❤r✐st✐❛♥ ❑✉❞❛❤❧ ✭■▼❆❉❆✮ ❖♥✲❧✐♥❡ ❈❤r♦♠❛t✐❝ ◆✉♠❜❡r ❏✉❧② ✼✱ ✷✵✶✹ ✾ ✴ ✷✵

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SLIDE 31

❚❤❡ ❖♥✲❧✐♥❡ ❈❤r♦♠❛t✐❝ ◆✉♠❜❡r

❖♥✲❧✐♥❡ ●r❛♣❤ ❝♦❧♦r✐♥❣ ❝❛♥ ❜❡ s❡❡♥ ❛s ❛ ❣❛♠❡ ❜❡t✇❡❡♥ t❤❡ ❉r❛✇❡r ❛♥❞ t❤❡ P❛✐♥t❡r✳ (G, k) ✐s t❤❡ ❣❛♠❡ ✐♥st❛♥❝❡ t❤❡② ❛❣r❡❡ ♦♥ ❜❡❢♦r❡ t❤❡ ❣❛♠❡✳ ❚❤❡ ❉r❛✇❡r ♣r❡s❡♥ts ❛ ✈❡rt❡① ♦♥ ❤✐s t✉r♥✳ ❚❤❡ P❛✐♥t❡r ❝♦❧♦rs t❤❛t ✈❡rt❡① ♦♥ ❤✐s t✉r♥✳ ❆t ❡❛❝❤ t✐♠❡✱ t❤❡ ♣r❡s❡♥t❡❞ ❣r❛♣❤ ✐s ✐s♦♠♦r♣❤✐❝ t♦ ❛♥ ✐♥❞✉❝❡❞ s✉❜❣r❛♣❤ ♦❢ ✳ ■❢ t❤❡ ♣❛✐♥t❡r ✐s ❛❜❧❡ t♦ ❝♦❧♦r t❤❡ ❡♥t✐r❡ ❣r❛♣❤ ✇✐t❤♦✉t ✉s✐♥❣ ♠♦r❡ t❤❛♥ ❝♦❧♦rs✱ ❤❡ ✇✐♥s✳ ❖t❤❡r✇✐s❡✱ ❤❡ ❧♦s❡s✳ ❚❤❡ ♣❛✐♥t❡r ❤❛s ❛ ✇✐♥♥✐♥❣ str❛t❡❣② ✐❢ ❛♥❞ ♦♥❧② ✐❢ ✳

❈❤r✐st✐❛♥ ❑✉❞❛❤❧ ✭■▼❆❉❆✮ ❖♥✲❧✐♥❡ ❈❤r♦♠❛t✐❝ ◆✉♠❜❡r ❏✉❧② ✼✱ ✷✵✶✹ ✾ ✴ ✷✵

slide-32
SLIDE 32

❚❤❡ ❖♥✲❧✐♥❡ ❈❤r♦♠❛t✐❝ ◆✉♠❜❡r

❖♥✲❧✐♥❡ ●r❛♣❤ ❝♦❧♦r✐♥❣ ❝❛♥ ❜❡ s❡❡♥ ❛s ❛ ❣❛♠❡ ❜❡t✇❡❡♥ t❤❡ ❉r❛✇❡r ❛♥❞ t❤❡ P❛✐♥t❡r✳ (G, k) ✐s t❤❡ ❣❛♠❡ ✐♥st❛♥❝❡ t❤❡② ❛❣r❡❡ ♦♥ ❜❡❢♦r❡ t❤❡ ❣❛♠❡✳ ❚❤❡ ❉r❛✇❡r ♣r❡s❡♥ts ❛ ✈❡rt❡① ♦♥ ❤✐s t✉r♥✳ ❚❤❡ P❛✐♥t❡r ❝♦❧♦rs t❤❛t ✈❡rt❡① ♦♥ ❤✐s t✉r♥✳ ❆t ❡❛❝❤ t✐♠❡✱ t❤❡ ♣r❡s❡♥t❡❞ ❣r❛♣❤ ✐s ✐s♦♠♦r♣❤✐❝ t♦ ❛♥ ✐♥❞✉❝❡❞ s✉❜❣r❛♣❤ ♦❢ G✳ ■❢ t❤❡ ♣❛✐♥t❡r ✐s ❛❜❧❡ t♦ ❝♦❧♦r t❤❡ ❡♥t✐r❡ ❣r❛♣❤ ✇✐t❤♦✉t ✉s✐♥❣ ♠♦r❡ t❤❛♥ ❝♦❧♦rs✱ ❤❡ ✇✐♥s✳ ❖t❤❡r✇✐s❡✱ ❤❡ ❧♦s❡s✳ ❚❤❡ ♣❛✐♥t❡r ❤❛s ❛ ✇✐♥♥✐♥❣ str❛t❡❣② ✐❢ ❛♥❞ ♦♥❧② ✐❢ ✳

❈❤r✐st✐❛♥ ❑✉❞❛❤❧ ✭■▼❆❉❆✮ ❖♥✲❧✐♥❡ ❈❤r♦♠❛t✐❝ ◆✉♠❜❡r ❏✉❧② ✼✱ ✷✵✶✹ ✾ ✴ ✷✵

slide-33
SLIDE 33

❚❤❡ ❖♥✲❧✐♥❡ ❈❤r♦♠❛t✐❝ ◆✉♠❜❡r

❖♥✲❧✐♥❡ ●r❛♣❤ ❝♦❧♦r✐♥❣ ❝❛♥ ❜❡ s❡❡♥ ❛s ❛ ❣❛♠❡ ❜❡t✇❡❡♥ t❤❡ ❉r❛✇❡r ❛♥❞ t❤❡ P❛✐♥t❡r✳ (G, k) ✐s t❤❡ ❣❛♠❡ ✐♥st❛♥❝❡ t❤❡② ❛❣r❡❡ ♦♥ ❜❡❢♦r❡ t❤❡ ❣❛♠❡✳ ❚❤❡ ❉r❛✇❡r ♣r❡s❡♥ts ❛ ✈❡rt❡① ♦♥ ❤✐s t✉r♥✳ ❚❤❡ P❛✐♥t❡r ❝♦❧♦rs t❤❛t ✈❡rt❡① ♦♥ ❤✐s t✉r♥✳ ❆t ❡❛❝❤ t✐♠❡✱ t❤❡ ♣r❡s❡♥t❡❞ ❣r❛♣❤ ✐s ✐s♦♠♦r♣❤✐❝ t♦ ❛♥ ✐♥❞✉❝❡❞ s✉❜❣r❛♣❤ ♦❢ G✳ ■❢ t❤❡ ♣❛✐♥t❡r ✐s ❛❜❧❡ t♦ ❝♦❧♦r t❤❡ ❡♥t✐r❡ ❣r❛♣❤ ✇✐t❤♦✉t ✉s✐♥❣ ♠♦r❡ t❤❛♥ k ❝♦❧♦rs✱ ❤❡ ✇✐♥s✳ ❖t❤❡r✇✐s❡✱ ❤❡ ❧♦s❡s✳ ❚❤❡ ♣❛✐♥t❡r ❤❛s ❛ ✇✐♥♥✐♥❣ str❛t❡❣② ✐❢ ❛♥❞ ♦♥❧② ✐❢ ✳

❈❤r✐st✐❛♥ ❑✉❞❛❤❧ ✭■▼❆❉❆✮ ❖♥✲❧✐♥❡ ❈❤r♦♠❛t✐❝ ◆✉♠❜❡r ❏✉❧② ✼✱ ✷✵✶✹ ✾ ✴ ✷✵

slide-34
SLIDE 34

❚❤❡ ❖♥✲❧✐♥❡ ❈❤r♦♠❛t✐❝ ◆✉♠❜❡r

❖♥✲❧✐♥❡ ●r❛♣❤ ❝♦❧♦r✐♥❣ ❝❛♥ ❜❡ s❡❡♥ ❛s ❛ ❣❛♠❡ ❜❡t✇❡❡♥ t❤❡ ❉r❛✇❡r ❛♥❞ t❤❡ P❛✐♥t❡r✳ (G, k) ✐s t❤❡ ❣❛♠❡ ✐♥st❛♥❝❡ t❤❡② ❛❣r❡❡ ♦♥ ❜❡❢♦r❡ t❤❡ ❣❛♠❡✳ ❚❤❡ ❉r❛✇❡r ♣r❡s❡♥ts ❛ ✈❡rt❡① ♦♥ ❤✐s t✉r♥✳ ❚❤❡ P❛✐♥t❡r ❝♦❧♦rs t❤❛t ✈❡rt❡① ♦♥ ❤✐s t✉r♥✳ ❆t ❡❛❝❤ t✐♠❡✱ t❤❡ ♣r❡s❡♥t❡❞ ❣r❛♣❤ ✐s ✐s♦♠♦r♣❤✐❝ t♦ ❛♥ ✐♥❞✉❝❡❞ s✉❜❣r❛♣❤ ♦❢ G✳ ■❢ t❤❡ ♣❛✐♥t❡r ✐s ❛❜❧❡ t♦ ❝♦❧♦r t❤❡ ❡♥t✐r❡ ❣r❛♣❤ ✇✐t❤♦✉t ✉s✐♥❣ ♠♦r❡ t❤❛♥ k ❝♦❧♦rs✱ ❤❡ ✇✐♥s✳ ❖t❤❡r✇✐s❡✱ ❤❡ ❧♦s❡s✳ ❚❤❡ ♣❛✐♥t❡r ❤❛s ❛ ✇✐♥♥✐♥❣ str❛t❡❣② ✐❢ ❛♥❞ ♦♥❧② ✐❢ χO(G) ≤ k✳

❈❤r✐st✐❛♥ ❑✉❞❛❤❧ ✭■▼❆❉❆✮ ❖♥✲❧✐♥❡ ❈❤r♦♠❛t✐❝ ◆✉♠❜❡r ❏✉❧② ✼✱ ✷✵✶✹ ✾ ✴ ✷✵

slide-35
SLIDE 35

❚❤❡ ❖♥✲❧✐♥❡ ❈❤r♦♠❛t✐❝ ◆✉♠❜❡r

❲❡ ❝❛♥ ❝♦♥s✐❞❡r t❤❡ ❢♦❧❧♦✇✐♥❣ r❡❧❛t❡❞ ♣r♦❜❧❡♠✿

❖♥✲❧✐♥❡ ❈❤r♦♠❛t✐❝ ◆✉♠❜❡r ✇✐t❤ Pr❡✲❝♦❧♦r✐♥❣

  • ✐✈❡♥ ❛ st❛t❡ ✐♥ t❤❡ ❖♥✲❧✐♥❡ ●r❛♣❤ ❈♦❧♦r✐♥❣ ❣❛♠❡✱ ❞♦❡s t❤❡ ♣❛✐♥t❡r ❤❛✈❡ ❛

✇✐♥♥✐♥❣ str❛t❡❣② ❢r♦♠ t❤❛t st❛t❡❄ ❆ st❛t❡ ✐♥ t❤❡ ❣❛♠❡ ♠❡❛♥s t❤❛t ❛ ♣❛rt ♦❢ t❤❡ ❣r❛♣❤ ❤❛s ❛❧r❡❛❞② ❜❡❡♥ r❡✈❡❛❧❡❞ ❛♥❞ ❝♦❧♦r✳ ❚❤✐s ♣r♦❜❧❡♠✱ ✇❤✐❝❤ ✐s ❛❧s♦ ❦♥♦✇♥ ❛s ❖♥✲❧✐♥❡ ❈❤r♦♠❛t✐❝ ◆✉♠❜❡r ✇✐t❤ Pr❡✲❝♦❧♦r✐♥❣✱ ✐s P❙P❆❈❊✲❈♦♠♣❧❡t❡✳

❈❤r✐st✐❛♥ ❑✉❞❛❤❧ ✭■▼❆❉❆✮ ❖♥✲❧✐♥❡ ❈❤r♦♠❛t✐❝ ◆✉♠❜❡r ❏✉❧② ✼✱ ✷✵✶✹ ✶✵ ✴ ✷✵

slide-36
SLIDE 36

❙❦❡t❝❤ ♦❢ Pr♦♦❢

❘❡❞✉❝t✐♦♥ ❢r♦♠ q✉❛♥t✐✜❡❞ s❛t✐s✜❛❜✐❧✐t② ✐♥ ✸✲❉◆❋✳ F = ∀x✶∃x✷∀x✸ : (x✶ ∧ x✷ ∧ x✸) ∨ (x✶ ∧ x✷ ∧ x✸) ∨ (x✶ ∧ x✷ ∧ x✸) ▼❛♣♣✐♥❣ F ✐♥t♦ ❣r❛♣❤ G ✇✐t❤ ♣r❡❝♦❧♦r✐♥❣ ❛♥❞ ❛ k✳ tr✉❡ P❛✐♥t❡r ❝❛♥ ❝♦❧♦r ✉s✐♥❣ ❛t ♠♦st ❝♦❧♦rs✳ ✐s ❢❛❧s❡ ❉r❛✇❡r ❝❛♥ ❢♦r❝❡ t❤❡ ♣❛✐♥t❡r t♦ ✉s❡ ♠♦r❡ t❤❛♥ ❝♦❧♦rs✳ ❚✇♦ ✈❡rt✐❝❡s ❢♦r ❡❛❝❤ ✈❛r✐❛❜❧❡✳ ❖♥❡ ✈❡rt❡① ❢♦r ❡❛❝❤ t❡r♠✳

❈❤r✐st✐❛♥ ❑✉❞❛❤❧ ✭■▼❆❉❆✮ ❖♥✲❧✐♥❡ ❈❤r♦♠❛t✐❝ ◆✉♠❜❡r ❏✉❧② ✼✱ ✷✵✶✹ ✶✶ ✴ ✷✵

slide-37
SLIDE 37

❙❦❡t❝❤ ♦❢ Pr♦♦❢

❘❡❞✉❝t✐♦♥ ❢r♦♠ q✉❛♥t✐✜❡❞ s❛t✐s✜❛❜✐❧✐t② ✐♥ ✸✲❉◆❋✳ F = ∀x✶∃x✷∀x✸ : (x✶ ∧ x✷ ∧ x✸) ∨ (x✶ ∧ x✷ ∧ x✸) ∨ (x✶ ∧ x✷ ∧ x✸) ▼❛♣♣✐♥❣ F ✐♥t♦ ❣r❛♣❤ G ✇✐t❤ ♣r❡❝♦❧♦r✐♥❣ ❛♥❞ ❛ k✳ F tr✉❡ ⇔ P❛✐♥t❡r ❝❛♥ ❝♦❧♦r G ✉s✐♥❣ ❛t ♠♦st k ❝♦❧♦rs✳ ✐s ❢❛❧s❡ ❉r❛✇❡r ❝❛♥ ❢♦r❝❡ t❤❡ ♣❛✐♥t❡r t♦ ✉s❡ ♠♦r❡ t❤❛♥ ❝♦❧♦rs✳ ❚✇♦ ✈❡rt✐❝❡s ❢♦r ❡❛❝❤ ✈❛r✐❛❜❧❡✳ ❖♥❡ ✈❡rt❡① ❢♦r ❡❛❝❤ t❡r♠✳

❈❤r✐st✐❛♥ ❑✉❞❛❤❧ ✭■▼❆❉❆✮ ❖♥✲❧✐♥❡ ❈❤r♦♠❛t✐❝ ◆✉♠❜❡r ❏✉❧② ✼✱ ✷✵✶✹ ✶✶ ✴ ✷✵

slide-38
SLIDE 38

❙❦❡t❝❤ ♦❢ Pr♦♦❢

❘❡❞✉❝t✐♦♥ ❢r♦♠ q✉❛♥t✐✜❡❞ s❛t✐s✜❛❜✐❧✐t② ✐♥ ✸✲❉◆❋✳ F = ∀x✶∃x✷∀x✸ : (x✶ ∧ x✷ ∧ x✸) ∨ (x✶ ∧ x✷ ∧ x✸) ∨ (x✶ ∧ x✷ ∧ x✸) ▼❛♣♣✐♥❣ F ✐♥t♦ ❣r❛♣❤ G ✇✐t❤ ♣r❡❝♦❧♦r✐♥❣ ❛♥❞ ❛ k✳ F tr✉❡ ⇔ P❛✐♥t❡r ❝❛♥ ❝♦❧♦r G ✉s✐♥❣ ❛t ♠♦st k ❝♦❧♦rs✳ F ✐s ❢❛❧s❡ ⇔ ❉r❛✇❡r ❝❛♥ ❢♦r❝❡ t❤❡ ♣❛✐♥t❡r t♦ ✉s❡ ♠♦r❡ t❤❛♥ k ❝♦❧♦rs✳ ❚✇♦ ✈❡rt✐❝❡s ❢♦r ❡❛❝❤ ✈❛r✐❛❜❧❡✳ ❖♥❡ ✈❡rt❡① ❢♦r ❡❛❝❤ t❡r♠✳

❈❤r✐st✐❛♥ ❑✉❞❛❤❧ ✭■▼❆❉❆✮ ❖♥✲❧✐♥❡ ❈❤r♦♠❛t✐❝ ◆✉♠❜❡r ❏✉❧② ✼✱ ✷✵✶✹ ✶✶ ✴ ✷✵

slide-39
SLIDE 39

❙❦❡t❝❤ ♦❢ Pr♦♦❢

❘❡❞✉❝t✐♦♥ ❢r♦♠ q✉❛♥t✐✜❡❞ s❛t✐s✜❛❜✐❧✐t② ✐♥ ✸✲❉◆❋✳ F = ∀x✶∃x✷∀x✸ : (x✶ ∧ x✷ ∧ x✸) ∨ (x✶ ∧ x✷ ∧ x✸) ∨ (x✶ ∧ x✷ ∧ x✸) ▼❛♣♣✐♥❣ F ✐♥t♦ ❣r❛♣❤ G ✇✐t❤ ♣r❡❝♦❧♦r✐♥❣ ❛♥❞ ❛ k✳ F tr✉❡ ⇔ P❛✐♥t❡r ❝❛♥ ❝♦❧♦r G ✉s✐♥❣ ❛t ♠♦st k ❝♦❧♦rs✳ F ✐s ❢❛❧s❡ ⇔ ❉r❛✇❡r ❝❛♥ ❢♦r❝❡ t❤❡ ♣❛✐♥t❡r t♦ ✉s❡ ♠♦r❡ t❤❛♥ k ❝♦❧♦rs✳ ❚✇♦ ✈❡rt✐❝❡s ❢♦r ❡❛❝❤ ✈❛r✐❛❜❧❡✳ ❖♥❡ ✈❡rt❡① ❢♦r ❡❛❝❤ t❡r♠✳

❈❤r✐st✐❛♥ ❑✉❞❛❤❧ ✭■▼❆❉❆✮ ❖♥✲❧✐♥❡ ❈❤r♦♠❛t✐❝ ◆✉♠❜❡r ❏✉❧② ✼✱ ✷✵✶✹ ✶✶ ✴ ✷✵

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SLIDE 40

❙❦❡t❝❤ ♦❢ Pr♦♦❢

❘❡❞✉❝t✐♦♥ ❢r♦♠ q✉❛♥t✐✜❡❞ s❛t✐s✜❛❜✐❧✐t② ✐♥ ✸✲❉◆❋✳ F = ∀x✶∃x✷∀x✸ : (x✶ ∧ x✷ ∧ x✸) ∨ (x✶ ∧ x✷ ∧ x✸) ∨ (x✶ ∧ x✷ ∧ x✸) ▼❛♣♣✐♥❣ F ✐♥t♦ ❣r❛♣❤ G ✇✐t❤ ♣r❡❝♦❧♦r✐♥❣ ❛♥❞ ❛ k✳ F tr✉❡ ⇔ P❛✐♥t❡r ❝❛♥ ❝♦❧♦r G ✉s✐♥❣ ❛t ♠♦st k ❝♦❧♦rs✳ F ✐s ❢❛❧s❡ ⇔ ❉r❛✇❡r ❝❛♥ ❢♦r❝❡ t❤❡ ♣❛✐♥t❡r t♦ ✉s❡ ♠♦r❡ t❤❛♥ k ❝♦❧♦rs✳ ❚✇♦ ✈❡rt✐❝❡s ❢♦r ❡❛❝❤ ✈❛r✐❛❜❧❡✳ ❖♥❡ ✈❡rt❡① ❢♦r ❡❛❝❤ t❡r♠✳

❈❤r✐st✐❛♥ ❑✉❞❛❤❧ ✭■▼❆❉❆✮ ❖♥✲❧✐♥❡ ❈❤r♦♠❛t✐❝ ◆✉♠❜❡r ❏✉❧② ✼✱ ✷✵✶✹ ✶✶ ✴ ✷✵

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SLIDE 41

❙❦❡t❝❤ ♦❢ Pr♦♦❢ v✶ v✶ v✷ v✷ v✸ v✸ C✶ C✷ C✸

F = ∀x✶∃x✷∀x✸ : (x✶ ∧ x✷ ∧ x✸) ∨ (x✶ ∧ x✷ ∧ x✸) ∨ (x✶ ∧ x✷ ∧ x✸)

❚❤✐s ✐s ❛ ♣❛rt ♦❢ t❤❡ ❣r❛♣❤✳

❈❤r✐st✐❛♥ ❑✉❞❛❤❧ ✭■▼❆❉❆✮ ❖♥✲❧✐♥❡ ❈❤r♦♠❛t✐❝ ◆✉♠❜❡r ❏✉❧② ✼✱ ✷✵✶✹ ✶✷ ✴ ✷✵

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SLIDE 42

❙❦❡t❝❤ ♦❢ Pr♦♦❢ v✶ v✶ v✷ v✷ v✸ v✸ C✶ C✷ C✸

F = ∀x✶∃x✷∀x✸ : (x✶ ∧ x✷ ∧ x✸) ∨ (x✶ ∧ x✷ ∧ x✸) ∨ (x✶ ∧ x✷ ∧ x✸)

❱❛r✐❛❜❧❡s ❝❛♥ ♦♥❧② ❣❡t ❝♦❧♦r ✬tr✉❡✬ ♦r ✬❢❛❧s❡✬ ✭✉s✐♥❣ ♣r❡✲❝♦❧♦r✐♥❣✮ ■❢ Ci ❣❡ts ❝♦❧♦r ✬❢❛❧s❡✬ ✭✇✐t❤♦✉t ❝♦♥✢✐❝t ✇✐t❤ ✈❛r✐❛❜❧❡s✮✱ ♣❛✐♥t❡r ✇✐♥s✳ ❖t❤❡r✇✐s❡✱ ❤❡ ❧♦s❡s✳

❈❤r✐st✐❛♥ ❑✉❞❛❤❧ ✭■▼❆❉❆✮ ❖♥✲❧✐♥❡ ❈❤r♦♠❛t✐❝ ◆✉♠❜❡r ❏✉❧② ✼✱ ✷✵✶✹ ✶✸ ✴ ✷✵

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SLIDE 43

❙❦❡t❝❤ ♦❢ Pr♦♦❢ v✶ v✶ v✷ v✷ v✸ v✸ C✶ C✷ C✸

F = ∀x✶∃x✷∀x✸ : (x✶ ∧ x✷ ∧ x✸) ∨ (x✶ ∧ x✷ ∧ x✸) ∨ (x✶ ∧ x✷ ∧ x✸)

❲❤❡♥ ❛ ✈❛r✐❛❜❧❡ ♦r ❝❧❛✉s❡ ✐s r❡q✉❡st❡❞✱ t❤❡ ♣❛✐♥t❡r ❦♥♦✇s t❤❡ ✐♥❞❡① ✭❜② ♣r❡✲❝♦❧♦r✐♥❣✮✳ ❋♦r t❤❡ ❡①✐st❡♥t✐❛❧❧② q✉❛♥t✐✜❡❞ ✈❛r✐❛❜❧❡s✱ t❤❡ ♣❛✐♥t❡r ❛❧s♦ ❦♥♦✇ ✐❢ ✐t ✐s vi ♦r vi✳

❈❤r✐st✐❛♥ ❑✉❞❛❤❧ ✭■▼❆❉❆✮ ❖♥✲❧✐♥❡ ❈❤r♦♠❛t✐❝ ◆✉♠❜❡r ❏✉❧② ✼✱ ✷✵✶✹ ✶✹ ✴ ✷✵

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SLIDE 44

❙❦❡t❝❤ ♦❢ Pr♦♦❢ v✶ v✶ v✷ v✷ v✸ v✸ C✶ C✷ C✸

F = ∀x✶∃x✷∀x✸ : (x✶ ∧ x✷ ∧ x✸) ∨ (x✶ ∧ x✷ ∧ x✸) ∨ (x✶ ∧ x✷ ∧ x✸)

■❢ t❤❡ ❢♦r♠✉❧❛ ✐s ❢❛❧s❡✱ ❚❤❡ ❉r❛✇❡r ❝❛♥ ✇✐♥✳ ❍❡ r❡q✉❡sts t❤❡ t✇♦ ✈❡rt✐❝❡s ❢♦r ❡❛❝❤ ✈❛r✐❛❜❧❡ ✐♥ ♦r❞❡r✳ ❋♦r t❤❡ ✉♥✐✈❡rs❛❧❧② q✉❛♥t✐✜❡❞ ✈❛r✐❛❜❧❡s✱ ❤❡ ❝❛♥ ❞❡❝✐❞❡ t❤❡ tr✉t❤ ❛ss✐❣♥♠❡♥t✳

❈❤r✐st✐❛♥ ❑✉❞❛❤❧ ✭■▼❆❉❆✮ ❖♥✲❧✐♥❡ ❈❤r♦♠❛t✐❝ ◆✉♠❜❡r ❏✉❧② ✼✱ ✷✵✶✹ ✶✺ ✴ ✷✵

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SLIDE 45

❙❦❡t❝❤ ♦❢ Pr♦♦❢ v✶ v✶

F = ∀x✶∃x✷∀x✸ : (x✶ ∧ x✷ ∧ x✸) ∨ (x✶ ∧ x✷ ∧ x✸) ∨ (x✶ ∧ x✷ ∧ x✸)

❈❤r✐st✐❛♥ ❑✉❞❛❤❧ ✭■▼❆❉❆✮ ❖♥✲❧✐♥❡ ❈❤r♦♠❛t✐❝ ◆✉♠❜❡r ❏✉❧② ✼✱ ✷✵✶✹ ✶✻ ✴ ✷✵

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SLIDE 46

❙❦❡t❝❤ ♦❢ Pr♦♦❢ v✶❂❚ v✶❂❋

F = ∀x✶∃x✷∀x✸ : (x✶ ∧ x✷ ∧ x✸) ∨ (x✶ ∧ x✷ ∧ x✸) ∨ (x✶ ∧ x✷ ∧ x✸)

❈❤r✐st✐❛♥ ❑✉❞❛❤❧ ✭■▼❆❉❆✮ ❖♥✲❧✐♥❡ ❈❤r♦♠❛t✐❝ ◆✉♠❜❡r ❏✉❧② ✼✱ ✷✵✶✹ ✶✼ ✴ ✷✵

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SLIDE 47

❙❦❡t❝❤ ♦❢ Pr♦♦❢ v ✶❂❚ v✶❂❋

F = ∀x✶∃x✷∀x✸ : (x✶ ∧ x✷ ∧ x✸) ∨ (x✶ ∧ x✷ ∧ x✸) ∨ (x✶ ∧ x✷ ∧ x✸)

❈❤r✐st✐❛♥ ❑✉❞❛❤❧ ✭■▼❆❉❆✮ ❖♥✲❧✐♥❡ ❈❤r♦♠❛t✐❝ ◆✉♠❜❡r ❏✉❧② ✼✱ ✷✵✶✹ ✶✼ ✴ ✷✵

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SLIDE 48

❙❦❡t❝❤ ♦❢ Pr♦♦❢ v ✶❂❚ v✶❂❋ v✷ v ✷

F = ∀x✶∃x✷∀x✸ : (x✶ ∧ x✷ ∧ x✸) ∨ (x✶ ∧ x✷ ∧ x✸) ∨ (x✶ ∧ x✷ ∧ x✸)

❈❤r✐st✐❛♥ ❑✉❞❛❤❧ ✭■▼❆❉❆✮ ❖♥✲❧✐♥❡ ❈❤r♦♠❛t✐❝ ◆✉♠❜❡r ❏✉❧② ✼✱ ✷✵✶✹ ✶✼ ✴ ✷✵

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SLIDE 49

❙❦❡t❝❤ ♦❢ Pr♦♦❢ v ✶❂❚ v✶❂❋ v✷❂❋ v ✷❂❚

F = ∀x✶∃x✷∀x✸ : (x✶ ∧ x✷ ∧ x✸) ∨ (x✶ ∧ x✷ ∧ x✸) ∨ (x✶ ∧ x✷ ∧ x✸)

❈❤r✐st✐❛♥ ❑✉❞❛❤❧ ✭■▼❆❉❆✮ ❖♥✲❧✐♥❡ ❈❤r♦♠❛t✐❝ ◆✉♠❜❡r ❏✉❧② ✼✱ ✷✵✶✹ ✶✼ ✴ ✷✵

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SLIDE 50

❙❦❡t❝❤ ♦❢ Pr♦♦❢ v ✶❂❚ v✶❂❋ v✷❂❋ v ✷❂❚ v✸ v✸

F = ∀x✶∃x✷∀x✸ : (x✶ ∧ x✷ ∧ x✸) ∨ (x✶ ∧ x✷ ∧ x✸) ∨ (x✶ ∧ x✷ ∧ x✸)

❈❤r✐st✐❛♥ ❑✉❞❛❤❧ ✭■▼❆❉❆✮ ❖♥✲❧✐♥❡ ❈❤r♦♠❛t✐❝ ◆✉♠❜❡r ❏✉❧② ✼✱ ✷✵✶✹ ✶✼ ✴ ✷✵

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SLIDE 51

❙❦❡t❝❤ ♦❢ Pr♦♦❢ v ✶❂❚ v✶❂❋ v✷❂❋ v ✷❂❚ v✸❂❚ v✸❂❋

F = ∀x✶∃x✷∀x✸ : (x✶ ∧ x✷ ∧ x✸) ∨ (x✶ ∧ x✷ ∧ x✸) ∨ (x✶ ∧ x✷ ∧ x✸)

❈❤r✐st✐❛♥ ❑✉❞❛❤❧ ✭■▼❆❉❆✮ ❖♥✲❧✐♥❡ ❈❤r♦♠❛t✐❝ ◆✉♠❜❡r ❏✉❧② ✼✱ ✷✵✶✹ ✶✼ ✴ ✷✵

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SLIDE 52

❙❦❡t❝❤ ♦❢ Pr♦♦❢ v ✶❂❚ v✶❂❋ v✷❂❋ v ✷❂❚ v✸❂❚ v ✸❂❋

F = ∀x✶∃x✷∀x✸ : (x✶ ∧ x✷ ∧ x✸) ∨ (x✶ ∧ x✷ ∧ x✸) ∨ (x✶ ∧ x✷ ∧ x✸)

❈❤r✐st✐❛♥ ❑✉❞❛❤❧ ✭■▼❆❉❆✮ ❖♥✲❧✐♥❡ ❈❤r♦♠❛t✐❝ ◆✉♠❜❡r ❏✉❧② ✼✱ ✷✵✶✹ ✶✼ ✴ ✷✵

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SLIDE 53

❙❦❡t❝❤ ♦❢ Pr♦♦❢ v ✶❂❚ v✶❂❋ v✷❂❋ v ✷❂❚ v✸❂❚ v ✸❂❋ C✶❂✶

F = ∀x✶∃x✷∀x✸ : (x✶ ∧ x✷ ∧ x✸) ∨ (x✶ ∧ x✷ ∧ x✸) ∨ (x✶ ∧ x✷ ∧ x✸)

❈❤r✐st✐❛♥ ❑✉❞❛❤❧ ✭■▼❆❉❆✮ ❖♥✲❧✐♥❡ ❈❤r♦♠❛t✐❝ ◆✉♠❜❡r ❏✉❧② ✼✱ ✷✵✶✹ ✶✼ ✴ ✷✵

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SLIDE 54

❙❦❡t❝❤ ♦❢ Pr♦♦❢ v ✶❂❚ v✶❂❋ v✷❂❋ v ✷❂❚ v✸❂❚ v ✸❂❋ C✶❂✶ C✷❂✷

F = ∀x✶∃x✷∀x✸ : (x✶ ∧ x✷ ∧ x✸) ∨ (x✶ ∧ x✷ ∧ x✸) ∨ (x✶ ∧ x✷ ∧ x✸)

❈❤r✐st✐❛♥ ❑✉❞❛❤❧ ✭■▼❆❉❆✮ ❖♥✲❧✐♥❡ ❈❤r♦♠❛t✐❝ ◆✉♠❜❡r ❏✉❧② ✼✱ ✷✵✶✹ ✶✼ ✴ ✷✵

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SLIDE 55

❙❦❡t❝❤ ♦❢ Pr♦♦❢ v ✶❂❚ v✶❂❋ v✷❂❋ v ✷❂❚ v✸❂❚ v ✸❂❋ C✶❂✶ C✷❂✷ C✸❂✸

F = ∀x✶∃x✷∀x✸ : (x✶ ∧ x✷ ∧ x✸) ∨ (x✶ ∧ x✷ ∧ x✸) ∨ (x✶ ∧ x✷ ∧ x✸)

■❢ ♦♥❡ ❝❧❛✉s❡ ❤❛❞ ❜❡❡♥ s❛t✐s✜❡❞✱ ✐t ❝♦✉❧❞ ❤❛✈❡ r❡❝❡✐✈❡❞ t❤❡ ❝♦❧♦r ❢❛❧s❡ ❛♥❞ t❤❡ ♣❛✐♥t❡r ✇♦✉❧❞ ❤❛✈❡ ✇♦♥✳

❈❤r✐st✐❛♥ ❑✉❞❛❤❧ ✭■▼❆❉❆✮ ❖♥✲❧✐♥❡ ❈❤r♦♠❛t✐❝ ◆✉♠❜❡r ❏✉❧② ✼✱ ✷✵✶✹ ✶✼ ✴ ✷✵

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SLIDE 56

❙❦❡t❝❤ ♦❢ Pr♦♦❢ v ✶❂❚ v✶❂❋ v✷❂❋ v ✷❂❚ v✸❂❚ v ✸❂❋ C✶❂✶ C✷❂✷ C✸❂✸

F = ∀x✶∃x✷∀x✸ : (x✶ ∧ x✷ ∧ x✸) ∨ (x✶ ∧ x✷ ∧ x✸) ∨ (x✶ ∧ x✷ ∧ x✸)

■❢ ♦♥❡ ❝❧❛✉s❡ ❤❛❞ ❜❡❡♥ s❛t✐s✜❡❞✱ ✐t ❝♦✉❧❞ ❤❛✈❡ r❡❝❡✐✈❡❞ t❤❡ ❝♦❧♦r ❢❛❧s❡ ❛♥❞ t❤❡ ♣❛✐♥t❡r ✇♦✉❧❞ ❤❛✈❡ ✇♦♥✳

❈❤r✐st✐❛♥ ❑✉❞❛❤❧ ✭■▼❆❉❆✮ ❖♥✲❧✐♥❡ ❈❤r♦♠❛t✐❝ ◆✉♠❜❡r ❏✉❧② ✼✱ ✷✵✶✹ ✶✼ ✴ ✷✵

slide-57
SLIDE 57

❙❦❡t❝❤ ♦❢ Pr♦♦❢ v✶ v✶ v✷ v✷ v✸ v✸ C✶ C✷ C✸

F = ∀x✶∃x✷∀x✸ : (x✶ ∧ x✷ ∧ x✸) ∨ (x✶ ∧ x✷ ∧ x✸) ∨ (x✶ ∧ x✷ ∧ x✸)

❆♥ ❛❞❞✐t✐♦♥❛❧ ❣❛❞❣❡t ❡♥s✉r❡s t❤❛t t❤❡ ❉r❛✇❡r ♠✉st r❡q✉❡st ✈❛r✐❛❜❧❡s ✐♥ ♦r❞❡r✳ ❲❤❡♥ F ✐s tr✉❡✱ t❤❡ P❛✐♥t❡r ❝❛♥ ❝♦❧♦r ❡①✐st❡♥t✐❛❧❧② q✉❛♥t✐✜❡❞ ✈❛r✐❛❜❧❡s ❛❝❝♦r❞✐♥❣ t♦ t❤❡ ✬❝♦rr❡❝t✬ tr✉t❤ ❛ss✐❣♥♠❡♥t ❢r♦♠ t❤❡ s❛t✐s✜❛❜✐❧✐t② ♣r♦❜❧❡♠✳ ❖♥❡ ❝❧❛✉s❡ ✇✐❧❧ ❤❛✈❡ ♥♦ ♥❡✐❣❤❜♦✉r ✇✐t❤ ❝♦❧♦r ❢❛❧s❡✱ s♦ ✐t ❝❛♥ ❣❡t t❤❛t ❝♦❧♦r✳

❈❤r✐st✐❛♥ ❑✉❞❛❤❧ ✭■▼❆❉❆✮ ❖♥✲❧✐♥❡ ❈❤r♦♠❛t✐❝ ◆✉♠❜❡r ❏✉❧② ✼✱ ✷✵✶✹ ✶✽ ✴ ✷✵

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SLIDE 58

❖t❤❡r r❡s✉❧ts✱ ❖♣❡♥ Pr♦❜❧❡♠s✱ ❛♥❞ ❈♦♥❥❡❝t✉r❡s

■t ✐s st✐❧❧ ✉♥❦♥♦✇♥ ✐❢ t❤❡ ♣r♦❜❧❡♠ r❡♠❛✐♥s P❙P❆❈❊✲❈♦♠♣❧❡t❡ ✇✐t❤♦✉t t❤❡ ♣r❡✲❝♦❧♦r✐♥❣✳

❚❤❡♦r❡♠

❖♥❧✐♥❡ ❈❤r♦♠❛t✐❝ ◆✉♠❜❡r ✇✐t❤♦✉t Pr❡✲❝♦❧♦r✐♥❣ ✐s ❝♦◆P✲❍❛r❞✳

❚❤❡♦r❡♠

❖♥❧✐♥❡ ❈❤r♦♠❛t✐❝ ◆✉♠❜❡r ✇✐t❤♦✉t Pr❡✲❝♦❧♦r✐♥❣ ✐s Σp

✷✲❍❛r❞ ✐♥ ♠✉❧t✐❣r❛♣❤s✳

❈♦♥❥❡❝t✉r❡ ❬●②❛r❢❛s✱ ▲❡❤❡❧✱ ❑✐r❛❧② ✾✽❪

  • ✐✈❡♥ ❛ ❣r❛♣❤✱ G✱ ✐t ✐s ◆P✲❤❛r❞ t♦ ❞❡❝✐❞❡ ✐❢ χO(G) ≤ ✹

❈❤r✐st✐❛♥ ❑✉❞❛❤❧ ✭■▼❆❉❆✮ ❖♥✲❧✐♥❡ ❈❤r♦♠❛t✐❝ ◆✉♠❜❡r ❏✉❧② ✼✱ ✷✵✶✹ ✶✾ ✴ ✷✵

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SLIDE 59

❚❤❛♥❦ ②♦✉ ❢♦r ❧✐st❡♥✐♥❣✦

❈❤r✐st✐❛♥ ❑✉❞❛❤❧ ✭■▼❆❉❆✮ ❖♥✲❧✐♥❡ ❈❤r♦♠❛t✐❝ ◆✉♠❜❡r ❏✉❧② ✼✱ ✷✵✶✹ ✷✵ ✴ ✷✵