●✐③❡❧❡ ❏✉st✐♥♦ ❉✐♥✐③ ▼❛rt✐♥s
●r❛❢♦s✿ ❞❡✜♥✐çõ❡s ❡❧❡♠❡♥t❛r❡s ❡ ♠ét♦❞♦
♣r♦❜❛❜✐❧íst✐❝♦
●✐③❡❧❡ ❏✉st✐♥♦ ❉✐♥✐③ ▼❛rt✐♥s
●r❛❢♦s✿ ❞❡✜♥✐çõ❡s ❡❧❡♠❡♥t❛r❡s ❡ ♠ét♦❞♦
♣r♦❜❛❜✐❧íst✐❝♦
❚r❛❜❛❧❤♦ ❞❡ ❈♦♥❝❧✉sã♦ ❞❡ ❈✉rs♦ ❛♣r❡s❡♥t❛❞♦ ❛♦ ❈♦r♣♦ ❉♦❝❡♥t❡ ❞♦ ❈✉rs♦ ❞❡ Pós✲●r❛❞✉❛çã♦ ❡♠ ▼❛t❡♠át✐❝❛ ❡♠ r❡❞❡ ◆❛❝✐♦♥❛❧ ✲ P❘❖❋▼❆❚ ✲ ❉▼ ✲ ❈❈❊◆ ✲ ❯❋P❇✱ ❝♦♠♦ r❡q✉✐s✐t♦ ♣❛r❝✐❛❧ ♣❛r❛ ♦❜t❡♥çã♦ ❞♦ tít✉❧♦ ❞❡ ▼❡str❡ ❡♠ ▼❛t❡♠át✐❝❛✳
❖r✐❡♥t❛❞♦r✿ Pr♦❢✳ ❉r✳ ❆❧❡①❛♥❞r❡ ❞❡ ❇✉st❛♠❛♥t❡ ❙✐♠❛s
M386g Martins, Gizele Justino Diniz.
Grafos: definições elementares e método probabilístico / Gizele Justino Diniz Martins.-- João Pessoa, 2015.
79f. : il.
Orientador: Alexandre de Bustamante Simas Dissertação (Mestrado) – UFPB/CCEN
1. Matemática. 2. Grafos. 3. Número cromático. 4. Número de Ramsey. 5. Método probabilístico.
❉❊❉■❈❆❚Ó❘■❆
❆●❘❆❉❊❈■▼❊◆❚❖❙
❆❣r❛❞❡ç♦ ♣♦r ❡st❡ ♠♦♠❡♥t♦ ú♥✐❝♦ ❝♦♠ ✐♠❡♥s❛ ❢❡❧✐❝✐❞❛❞❡ ❡ s❛t✐s❢❛çã♦✳
❊♠ ♣r✐♠❡✐r♦ ❧✉❣❛r✱ ❛ ❉❡✉s✱ ❙❡♥❤♦r ❞❛ ♠✐♥❤❛ ✈✐❞❛✱ q✉❡ ♠❡ ❣✉✐♦✉✱ ✐❧✉♠✐♥❛♥❞♦ ♠❡✉ ❝❛♠✐♥❤♦✱ ❝♦♥❝❡❞❡♥❞♦✲♠❡ ❛ ❣r❛ç❛ ❞❡ ❝✉rs❛r ❡ss❡ ♠❡str❛❞♦✱ ❝♦♠ s✉❛ ❜❡♥çã♦ ❡♠ ❝❛❞❛ s❡♠❡str❡ ❡ ♥❛ ❡❧❛❜♦r❛çã♦ ❞♦ ♣r❡s❡♥t❡ tr❛❜❛❧❤♦✳
❆♦ ♠❡✉ ❡s♣♦s♦✱ ▼ár✐♦ ✖ ♣r❡s❡♥t❡ ❞❡ ❉❡✉s ❡♠ ♠✐♥❤❛ ✈✐❞❛ ✖ q✉❡ s❡♠♣r❡ ♠❡ ✐♥❝❡♥t✐✈♦✉✱ ❢❛③❡♥❞♦ ❝♦♠ q✉❡ ❛❝r❡❞✐t❛ss❡ ❡♠ ♠✐♠ ♠❡s♠❛✱ ❡♠ ♠✐♥❤❛ ❝❛♣❛❝✐❞❛❞❡✱ ❡ q✉❡ ❡✉ ✈✐s❧✉♠❜r❛ss❡ ❛❧é♠ ❞♦ ❤♦r✐③♦♥t❡✳ ❚❛♠❜é♠✱ ♣❡❧❛ s✉❛ ♣❛r❝❡r✐❛ ❡ ❝♦♠♣r❡❡♥sã♦ ❡♥q✉❛♥t♦ ❝♦♠♣❛♥❤❡✐r♦ ❡ ❝♦❧❡❣❛ ❞❡ ❡st✉❞♦✱ ♣r♦❢❡r✐♥❞♦ ❝♦♥s✐❞❡r❛çõ❡s ❡ ❝rít✐❝❛s ♥♦ ❞❡❝♦rr❡r ❞♦ ❝✉rs♦✳ ❆♦s ♠❡✉s ♣❛✐s✱ ●♦r❡tt✐ ❡ ▲✉✐③✱ q✉❡ ♠❡ ❛❝♦♠♣❛♥❤❛r❛♠ ❡ ✐♥❝❡♥t✐✈❛r❛♠ ❡♠ t♦❞♦s ♦s ♠♦♠❡♥t♦s✱ ❛✐♥❞❛ q✉❡ s❡♥t✐ss❡♠ ❛ ♠✐♥❤❛ ❛✉sê♥❝✐❛✳
❆♦ ♠❡✉ ✜❧❤♦ ❘❛❢❛❡❧ ❡ ❛s ♠✐♥❤❛s ✐r♠ãs✱ ●✐❧♠❛r❛✱ P❛✉❧❛ ❘♦❜❡rt❛ ❡ ❏❛q✉❡❧✐♥❡✱ q✉❡✱ ♣♦r ♠❡ ❛♠❛r❡♠✱ ❢♦r❛♠ r❛③ã♦ ❞❡ ❢♦rç❛ ❡ ✐♥s♣✐r❛çã♦ ❞✉r❛♥t❡ ❡ss❡s ❞♦✐s ❛♥♦s✳
❆s ♠✐♥❤❛s ♣r✐♠❛s ❏♦❝❡❧♠❛✱ ❇r✉♥❛ ❡ ❆❧✐♥❡✱ q✉❡ ❝♦♥t✐♥✉❛♠❡♥t❡ ♠❡ ❡♥❝♦r❛❥❛r❛♠ ❛ s❡❣✉✐r ❛❞✐❛♥t❡ ❡♠ ♣r♦❧ ❞♦ ♠❡✉ s♦♥❤♦✳
❆♦ ♠❡✉ ♦r✐❡♥t❛❞♦r✱ ❉r✳ ❆❧❡①❛♥❞r❡ ❇✉st❛♠❛♥t❡✱ ♣❡❧♦s ❡♥s✐♥❛♠❡♥t♦s ❡ ♣♦r t♦❞❛ ♣❛❝✐ê♥❝✐❛ ❞✉r❛♥t❡ ♦ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ❞❡st❛ ♣❡sq✉✐s❛✱ ❡♥r✐q✉❡❝❡♥❞♦✲❛ ❝♦♠ s❡✉ ❡①tr❛♦r❞✐♥ár✐♦ s❛❜❡r✳
❆ t♦❞♦s ♦s ♣r♦❢❡ss♦r❡s q✉❡ ❧❡❝✐♦♥❛r❛♠ ❛s ❞✐s❝✐♣❧✐♥❛s✱ ❡♠ ❡s♣❡❝✐❛❧ ❛ ❉r✳ ❇r✉♥♦ ❘✐❜❡✐r♦ ✖ ❝♦♦r❞❡♥❛❞♦r ❞♦ ❝✉rs♦ ❡ ♠❡♠❜r♦ ❞❡st❛ ❜❛♥❝❛ ✖✱ ❉r❛✳ ❊❧✐s❛♥❞r❛ ●❧♦ss✱ ▼s✳ ●✐❧♠❛r ❈♦rr❡✐❛✱ ❉r✳ ▲❡♥✐♠❛r ❆♥❞r❛❞❡✱ ❉r✳ ◆❛♣♦❧❡ó♥ ❚✉❡st❛✱ ♣❡❧❛ ❛t❡♥çã♦✱ ❛ss✐stê♥❝✐❛ ❡ ♣❡❧♦s ♣r♦✈❡✐t♦s♦s ❡♥s✐♥❛♠❡♥t♦s q✉❡ ♠❡ ♣r♦♣✐❝✐❛r❛♠✳ ❆✐♥❞❛✱ ❡①♣r❡ss♦ ♠✐♥❤❛ ❣r❛t✐❞ã♦ ❛♦ ♣r♦❢✳ ❉r✳ ❍❡♥r✐q✉❡ ❱✐tór✐♦ ♣♦r t❡r ❛❝❡✐t❛❞♦ ♦ ❝♦♥✈✐t❡ ♣❛r❛ ❝♦♠♣♦r ❡st❛ ✐❧✉str❡ ❜❛♥❝❛✳
❆s ♠✐♥❤❛s ❛♠✐❣❛s✱ ▼❛r✐❛ ❞♦ ❇♦♠ ❝♦♥s❡❧❤♦ ✖ ❝♦♠♣❛♥❤❡✐r❛ ❞❡ ♠❡str❛❞♦ ❡ ❞❡ ♠❛❞r✉❣❛❞❛s ❞❡ ❡st✉❞♦ ✖✱ ▲②❝②❛ ❈♦❡❧❤♦✱ ❑❛r✐♥❛ ❈❛♠✐❧♦✱ ■✈♦♥❡t❡ ❇❛r❜♦s❛✱ ❆♥❛ ▲ú❝✐❛ ▲❡✐t❡ ❡ ▲✐❧✐❛♥❡ ❇r❛❣❛ q✉❡ ❢♦r❛♠ ♠✉✐t♦ ✐♠♣♦rt❛♥t❡s ❛♦ t❡r♠♦ ❞❡st❛ ❢♦r♠❛çã♦ ❛❝❛❞ê♠✐❝❛✳
❚❛❧✈❡③ ♥ã♦ t❡♥❤❛ ❝♦♥s❡❣✉✐❞♦ ❢❛③❡r ♦ ♠❡❧❤♦r✱ ♠❛s ❧✉t❡✐ ♣❛r❛ q✉❡ ♦ ♠❡❧❤♦r
❢♦ss❡ ❢❡✐t♦✳ ◆ã♦ s♦✉ ♦ q✉❡ ❞❡✈❡r✐❛ s❡r✱ ♠❛s ●r❛ç❛s ❛ ❉❡✉s✱ ♥ã♦ s♦✉ ♦
q✉❡ ❡r❛ ❛♥t❡s✳
❘❊❙❯▼❖
◆❡st❡ tr❛❜❛❧❤♦✱ ❡st✉❞❛♠♦s ❛ t❡♦r✐❛ ❞♦s ❣r❛❢♦s✱ q✉❡ ✖ ❡♠❜♦r❛ s❡❥❛ ✉♠ ❝♦♥t❡ú❞♦ ♣♦✉❝♦ ❞✐❢✉♥❞✐❞♦✱ ✐♥❝❧✉s✐✈❡ ❡♠ â♠❜✐t♦ ❛❝❛❞ê♠✐❝♦ ✖ é ❞❡ ❡①tr❡♠❛ ✐♠♣♦rtâ♥❝✐❛ ♣❛r❛ ❛ r❡s♦❧✉çã♦ ❞❡ ✐♥ú♠❡r♦s ♣r♦❜❧❡♠❛s ♠❛t❡♠át✐❝♦s ❡ ♠♦❞❡❧♦s ❢ís✐❝♦s✳ ❆❧é♠ ❞✐ss♦✱ ❡ss❛ t❡♠át✐❝❛ ♣♦❞❡ s❡r ✈❡r✐✜❝❛❞❛ ❡♠ ❛♣❧✐❝❛çõ❡s ♥❛s ♠❛✐s ❞✐✈❡rs❛s ár❡❛s✱ ❡♥tr❡ ❛s q✉❛✐s ❝✐t❛♠♦s✿ ❝♦♠♣✉t❛❝✐♦♥❛❧✱ ❡❧étr✐❝❛✱ ❣❡♥ét✐❝❛✳ ❆❞♦t❛♠♦s ❛ ✐♥✈❡st✐❣❛çã♦ ❜✐❜❧✐♦❣rá✜❝❛ ❡ ❛ ♣❡sq✉✐s❛ ❡①♣❧♦r❛tór✐❛ ♣❛r❛ tr❛t❛r ♦ t❡♠❛ ❡♠ q✉❡stã♦✱ ♣r♦❝✉r❛♥❞♦ ❞❡✜♥✐r ❡ ❡s❝❧❛r❡❝❡r ❛ t❡♦r✐❛ s♦❜r❡❞✐t❛✱ ❝♦♠♦ t❛♠❜é♠ ❝♦♥tr✐❜✉✐r ❡♠ s✉❛ ❞✐❢✉sã♦✱ ♦ q✉❡ ♣♦ss✐❜✐❧✐t❛ ❛♦s ✐♥t❡❣r❛♥t❡s ❞❛ ❡❞✉❝❛çã♦ ❜ás✐❝❛ ❡ s✉♣❡r✐♦r t❡r❡♠ ✉♠ ❝♦♥t❛t♦ ❝♦♠ tã♦ ✐♠♣♦rt❛♥t❡ ❡ ❢❡❝✉♥❞♦ s❛❜❡r✱ ✉♠❛ ✈❡③ q✉❡ ❝♦♥s✐❞❡r❛çõ❡s ❡❧❡♠❡♥t❛r❡s s♦❜r❡ ❣r❛❢♦s ♥♦s ❛♣r♦①✐♠❛ ❞❛ ♣❡sq✉✐s❛ ❝✐❡♥tí✜❝❛✳ ■♥t❡r❝❛❧❛♠♦s ♦s ❝♦♥❝❡✐t♦s ❡ ❛s ❞❡♠♦♥str❛çõ❡s ❞❡ ❧❡♠❛s ❡ t❡♦r❡♠❛s ❝♦♠ ❛ ❛♣r❡s❡♥t❛çã♦ ❡ r❡s♦❧✉çã♦ ❞❡ ♣r♦❜❧❡♠❛s✳ ❊♠♣r❡❣❛♠♦s ✉♠❛ ❧✐♥❣✉❛❣❡♠ s✐♠♣❧❡s✱ ❞❡ ❢♦r♠❛ q✉❡ ✉♠ ❛❧✉♥♦ ❞♦ ❡♥s✐♥♦ ♠é❞✐♦ ♣♦ss❛ ❝♦♠♣r❡❡♥❞❡r✱ s❡♠✱ ♥♦ ❡♥t❛♥t♦✱ ♥♦s ❞✐st❛♥❝✐❛r ❞♦ r✐❣♦r ♠❛t❡♠át✐❝♦✳ ❊♠ t❡♠♣♦✱ ❛♣r❡s❡♥t❛♠♦s ♦ t❡♦r❡♠❛ ❞❛s q✉❛tr♦ ❝♦r❡s ❡ ♦ ♥ú♠❡r♦ ❞❡ ❘❛♠s❡②✱ ❝♦♠ ❞❡♠♦♥str❛çõ❡s ❞❡t❛❧❤❛❞❛s ❞❡st❡ ú❧t✐♠♦ r❡s✉❧t❛❞♦✳ P♦r ✜♠✱ ✉t✐❧✐③❛♠♦s ❝♦♥❝❡✐t♦s ❡ r❡s✉❧t❛❞♦s ♣✉r❛♠❡♥t❡ ❞❡ ❝♦♠❜✐♥❛tór✐❛ ❡ ♣r♦❜❛❜✐❧✐❞❛❞❡✱ ✉t✐❧✐③❛♥❞♦ ♦ ♠ét♦❞♦ ♣r♦❜❛❜✐❧íst✐❝♦ ♣❛r❛ ♣r♦✈❛r ❛ ❡①✐stê♥❝✐❛ ❞❡ ❣r❛❢♦s ❝♦♠ ❞❡t❡r♠✐♥❛❞❛s ♣r♦♣r✐❡❞❛❞❡s ✖ q✉❡ sã♦ ❞❡ ❞✐❢í❝✐❧ ❝♦♥str✉çã♦ ✖ ❡✱ ♣♦r ♠❡✐♦ ❞❡st❛ ❝♦♠♣r♦✈❛çã♦✱ ❝❤❡❣❛r ❛ ♦✉tr♦ ❣r❛❢♦ ❞❡s❡❥❛❞♦✳
❆❇❙❚❘❆❈❚
■♥ t❤✐s ✇♦r❦ ✇❡ st✉❞② t❤❡ ❣r❛♣❤ t❤❡♦r②✱ ✇❤✐❝❤ ✖ ❛❧t❤♦✉❣❤ ✐t ✐s s♦♠❡✇❤❛t ✇✐❞❡s♣r❡❛❞ ❝♦♥t❡♥t✱ ✐♥❝❧✉❞✐♥❣ ❛❝❛❞❡♠✐❛ ✖ ✐s ❡①tr❡♠❡❧② ✐♠♣♦rt❛♥t ❢♦r s♦❧✈✐♥❣ ♠❛♥② ♠❛t❤❡♠❛t✐❝❛❧ ♣r♦❜❧❡♠s ❛♥❞ ♣❤②s✐❝❛❧ ♠♦❞❡❧s✳ ▼♦r❡♦✈❡r✱ t❤✐s t❤❡♠❡ ❝❛♥ ❜❡ ❢♦✉♥❞ ✐♥ ❛♣♣❧✐❝❛t✐♦♥s ✐♥ s❡✈❡r❛❧ ❛r❡❛s✱ ✐♥❝❧✉❞✐♥❣ q✉♦t❡✿ ❝♦♠♣✉t❡r✱ ❡❧❡❝tr✐❝❛❧✱ ❣❡♥❡t✐❝✳ ❲❡ ❛❞♦♣t t❤❡ ❜✐❜❧✐♦❣r❛♣❤✐❝ r❡s❡❛r❝❤ ❛♥❞ ❡①♣❧♦r❛t♦r② r❡s❡❛r❝❤ t♦ ❞❡❛❧ ✇✐t❤ t❤❡ ✐ss✉❡ ❛t ❤❛♥❞✱ tr②✐♥❣ t♦ ❞❡✜♥❡ ❛♥❞ ❝❧❛r✐❢② t❤❡ ❛❢♦r❡s❛✐❞ t❤❡♦r②✱ ❜✉t ❛❧s♦ ❝♦♥tr✐❜✉t❡ t♦ ✐ts s♣r❡❛❞✱ ✇❤✐❝❤ ❡♥❛❜❧❡s ♠❡♠❜❡rs ♦❢ t❤❡ ❜❛s✐❝ ❛♥❞ ❤✐❣❤❡r ❡❞✉❝❛t✐♦♥ ❤❛✈❡ ❛ ❝♦♥t❛❝t ✇✐t❤ s✉❝❤ ❛♥ ✐♠♣♦rt❛♥t ❛♥❞ ❢r✉✐t❢✉❧ ❦♥♦✇✱ s✐♥❝❡ ❡❧❡♠❡♥t❛r② ❝♦♥s✐❞❡r❛t✐♦♥s ❣r❛♣❤s ❜r✐♥❣s ✉s ❝❧♦s❡r t♦ s❝✐❡♥t✐✜❝ r❡s❡❛r❝❤✳ ❲❡ ❛❧t❡r♥❛t❡ ❝♦♥❝❡♣ts ❛♥❞ st❛t❡♠❡♥ts ♦❢ ❧❡♠♠❛s ❛♥❞ t❤❡♦r❡♠s t♦ s♦❧✈❡ ♣r♦❜❧❡♠s✳ ❲❡ ✉s❡ s✐♠♣❧❡ ❧❛♥❣✉❛❣❡✱ s♦ t❤❛t ❛ ❤✐❣❤ s❝❤♦♦❧ st✉❞❡♥t ❝❛♥ ✉♥❞❡rst❛♥❞✱ ✇✐t❤♦✉t✱ ❤♦✇❡✈❡r✱ ❞✐st❛♥❝✐♥❣ ✉s ❢r♦♠ ♠❛t❤❡♠❛t✐❝❛❧ r✐❣♦r✳ ■♥ t✐♠❡✱ ✇❡ ♣r❡s❡♥t t❤❡ ❢♦✉r ❝♦❧♦r t❤❡♦r❡♠ t❤❡ ♥✉♠❜❡r ♦❢ ❘❛♠s❡②✱ ✇✐t❤ ❞❡t❛✐❧❡❞ st❛t❡♠❡♥ts ♦❢ t❤❡ ❧❛tt❡r r❡s✉❧t✳ ❋✐♥❛❧❧②✱ ✇❡ ✉s❡ ❝♦♥❝❡♣ts ❛♥❞ ♣✉r❡❧② ❝♦♠❜✐♥❛t♦r✐❛❧ r❡s✉❧ts ❛♥❞ ♣r♦❜❛❜✐❧✐t②✱ ✉s✐♥❣ t❤❡ ♣r♦❜❛❜✐❧✐st✐❝ ♠❡t❤♦❞ t♦ ♣r♦✈❡ t❤❡ ❡①✐st❡♥❝❡ ♦❢ ❣r❛♣❤s ✇✐t❤ ❝❡rt❛✐♥ ♣r♦♣❡rt✐❡s t❤❛t ❛r❡ ❞✐✣❝✉❧t ❝♦♥str✉❝t✐♦♥ ❛♥❞✱ t❤r♦✉❣❤ t❤✐s ❡✈✐❞❡♥❝❡✱ ❣❡t ♦t❤❡r ❞❡s✐r❡❞ ❣r❛♣❤✳
▲■❙❚❆ ❉❊ ❋■●❯❘❆❙
✶✳✶ ❊①❡♠♣❧♦s ❞❡ ●r❛❢♦s✿ ✭❛✮ ❣r❛❢♦ tr✐✈✐❛❧✱ ✭❜✮ ❣r❛❢♦ ❝♦♠♣❧❡t♦ ❡ ✭❝✮ ♣s❡✉❞♦❣r❛❢♦✳ ✶✻ ✶✳✷ ●r❛❢♦ ❞♦ t❛❜✉❧❡✐r♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✼ ✶✳✸ ●r❛❢♦ ♣r♦❜❧❡♠❛ ✷ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✽ ✶✳✹ ❊①❡♠♣❧✐✜❝❛♥❞♦ ❣r❛✉ ❞♦s ✈ért✐❝❡s ❡ ❣r❛❢♦ r✲r❡❣✉❧❛r ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✾ ✶✳✺ ●r❛❢♦ ❞♦ ❡①❡♠♣❧♦ ✼ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✶ ✶✳✻ ❈✐❝❧♦s ❡ tr✐❧❤❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✸ ✶✳✼ P♦♥t❡s ❞❡ ❑ö♥✐❣s❜❡r❣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✺ ✶✳✽ ●r❛❢♦ ❞❛s ♣♦♥t❡s ❞❞❡ ❑ö✐♥✐❣s❜❡r❣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✺ ✶✳✾ ●r❛❢♦s ✶✱ ✷ ❡ ✸ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✻ ✶✳✶✵ ●r❛❢♦s ✐s♦♠♦r❢♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✼ ✶✳✶✶ ●r❛❢♦ ♣r♦❜❧❡♠❛ ✶✶ ❜ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✾ ✶✳✶✷ ●r❛❢♦ ♣r♦❜❧❡♠❛ ✶✶❝ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✾ ✶✳✶✸ ●r❛❢♦ ❜✐♣❛rt✐❞♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✵ ✶✳✶✹ ❊①❡♠♣❧♦ ❞❡ ❣r❛❢♦ ❜✐♣❛rt✐❞♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✵ ✶✳✶✺ ●r❛❢♦ ✐s♦♠♦r❢♦ ❛♦ ❞❛ ✐❧✉str❛çã♦ ✶✳✶✹ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✶ ✶✳✶✻ ●r❛❢♦ ❜✐♣❛rt✐❞♦ ❝♦♠♣❧❡t♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✶ ✶✳✶✼ ➪r✈♦r❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✶ ✶✳✶✽ ❊①❡♠♣❧♦s ❞❡ ●r❛❢♦s✿ ✭❛✮ ❣r❛❢♦ ● ✱ ✭❜✮ ❍ s✉❜❣r❛❢♦ ❣❡r❛❞♦r ❞❡ ● ❡ ✭❝✮ ❏
✷✳✶ ●r❛❢♦ ❞❛s ♣r♦✈❛s ✜♥❛✐s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✺ ✷✳✷ ●r❛❢♦ ❞♦s ❡①❛♠❡s ✜♥❛✐s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✺ ✷✳✸ ❖✉tr❛ r❡♣r❡s❡♥t❛çã♦ ❞❛s ✜♥❛✐s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✻ ✷✳✹ ▼❛♣❛ ❞♦ ❇r❛s✐❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✻ ✷✳✺ ●r❛❢♦ r❡❧❛❝✐♦♥❛❞♦ ❛♦ ♠❛♣❛ ❞♦ ❇r❛s✐❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✼ ✷✳✻ ❈♦❧♦r❛çã♦ ❞❡ ❛r❡st❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✽ ✷✳✼ ●r❛❢♦ r❡❧❛t✐✈♦ ❛♦ ❡①❡♠♣❧♦ ✶✹ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✾ ✷✳✽ ❈♦♥t✐♥✉❛çã♦ ❞❛ ❝♦❧♦r❛çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✾ ✷✳✾ ●r❛❢♦ ❝♦♠♣❧❡t♦ K5 ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✵ ✷✳✶✵ ❘❛♠s❡② ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✷ ✷✳✶✶ ❊r❞♦s✱ ✶✾✾✷ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✸ ✸✳✶ ●r❛❢♦ ❝♦♠ χ= 4 ❡ ❣❂✹ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✽
❙❯▼➪❘■❖
■♥tr♦❞✉çã♦ ✶✷
✶ ❚❊❖❘■❆ ❉❖❙ ●❘❆❋❖❙ ✶✺
✶✳✶ ●r❛✉ ❞♦ ✈ért✐❝❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✽ ✶✳✷ ●r❛❢♦s ❝♦♥❡①♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✷ ✶✳✷✳✶ ●r❛❢♦ ❊✉❧❡r✐❛♥♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✹ ✶✳✸ ■s♦♠♦r✜s♠♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✼ ✶✳✹ ❉♦✐s t✐♣♦s ❡s♣❡❝✐❛✐s ❞❡ ❣r❛❢♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✵ ✶✳✹✳✶ ●r❛❢♦ ❜✐♣❛rt✐❞♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✵ ✶✳✹✳✷ ➪r✈♦r❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✶ ✶✳✺ ❙✉❜❣r❛❢♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✷ ✶✳✻ ●r❛❢♦ ♣❧❛♥❛r ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✸
✷ ❈❖▲❖❘❆➬➹❖ ✸✹
✷✳✶ ❈♦❧♦r✐♥❞♦ ❣r❛❢♦s ♣❧❛♥❛r❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✻ ✷✳✶✳✶ ❯♠ ♣♦✉❝♦ ❞❡ ❤✐stór✐❛✿ ◆ú♠❡r♦ ❝r♦♠át✐❝♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✼ ✷✳✷ ◆ú♠❡r♦ ❞❡ ❘❛♠s❡② ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✽ ✷✳✷✳✶ ❯♠ ♣♦✉❝♦ ❞❡ ❤✐stór✐❛✿ ❚❡♦r✐❛ ❞❡ ❘❛♠s❡② ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✷
✸ ▼➱❚❖❉❖ P❘❖❇❆❇■▲❮❙❚■❈❖ ✹✹
✸✳✷✳✶ ❙♦♠❛ ❞❡ ✈❛r✐á✈❡✐s ❛❧❡❛tór✐❛s ❡ ❡s♣❡r❛♥ç❛ ❞❛ s♦♠❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✾ ✸✳✸ ❈♦♥str✉çã♦ ❞❡ ❣r❛❢♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✼
✹ ❈❖◆❙■❉❊❘❆➬Õ❊❙ ❋■◆❆■❙ ✼✶
❘❊❋❊❘✃◆❈■❆❙ ✼✸
■◆❚❘❖❉❯➬➹❖
Pr✐♠❡✐r❛♠❡♥t❡✱ ❛♣♦♥t❛♠♦s q✉❡ ❛ t❡♦r✐❛ ❞♦s ●r❛❢♦s ❣❛♥❤♦✉ ❞❡st❛q✉❡ ♥❛ ♠❛t❡♠át✐❝❛ ❥á ♥♦ sé❝✉❧♦ ❳❳ ✭r❡❢❡rê♥❝✐❛ ❬✶✻❪✮✳ ❙❡♥❞♦✱ ❞❡st❛ ❢❡✐t❛✱ ✉♠❛ t❡♦r✐❛ ♥♦✈❛ ❡ ♣♦✉❝♦ ❝♦♥❤❡❝✐❞❛✱ ♠❛s q✉❡ t❡♠ ❣❛♥❤❛❞♦ ✉♠ ❛♠♣❧♦ ❡s♣❛ç♦ ❡♠ ✈✐rt✉❞❡ ❞❡ s✉❛s ❛♣❧✐❝❛çõ❡s ♥♦s ❞✐✈❡rs♦s ❝❛♠♣♦s ❞❛ ♠❛t❡♠át✐❝❛ ❡ ♥❛s ♦✉tr❛s ❝✐ê♥❝✐❛s✱ ❛ ❡①❡♠♣❧♦ ❞❛s ❡♥❣❡♥❤❛r✐❛s ❡ ❚■❈✬s✶✳
❯♠ ❣r❛❢♦ ♣♦❞❡ s❡r s✐♠♣❧❡s♠❡♥t❡ ❞❡s❝r✐t♦ ❝♦♠♦ ✉♠ ❝♦♥❥✉♥t♦ ❞❡ ♦❜❥❡t♦s q✉❡ s❡ ❝♦♥❡❝t❛ ❡♠ ♣❛r❡s✳ ❖s ❡❧❡♠❡♥t♦s ✭♦❜❥❡t♦s✮ ❡①✐st❡♥t❡s ♥❡ss❡ ❝♦♥❥✉♥t♦ sã♦ ❝❤❛♠❛❞♦s ❞❡ ✈ért✐❝❡s ❞♦ ❣r❛❢♦✳ ❆s ❧✐❣❛çõ❡s ✭r❡❧❛❝✐♦♥❛♠❡♥t♦s✮ ❡♥tr❡ ♦s ❡❧❡♠❡♥t♦s sã♦ ❝❤❛♠❛❞♦s ❞❡ ❛r❡st❛s ❞♦ ❣r❛❢♦✳ P❡r❝❡❜❡♥❞♦ q✉❡ ♦s ❝♦♠♣♦♥❡♥t❡s ❞♦s ❣r❛❢♦s ♣♦❞❡♠ ❡stá ✐♠❜r✐❝❛❞♦s ✉♥s ❝♦♠ ♦s ♦✉tr♦s✱ t♦r♥❛✲s❡ ❡✈✐❞❡♥t❡ s✉❛ s✐❣♥✐✜❝â♥❝✐❛ ❡ ✉t✐❧✐❞❛❞❡ ♥❛s ár❡❛s ❞❡ t❡❧❡❝♦♠✉♥✐❝❛çõ❡s✱ q✉í♠✐❝❛✱ ❣❡♥ét✐❝❛✱ ❧♦❣íst✐❝❛✱ ✐♥❢♦r♠át✐❝❛✱ ❡♥❣❡♥❤❛r✐❛ ❡❧étr✐❝❛✱ r❡❞❡s ❞❡ ❝♦♠♣✉t❛❞♦r❡s✱ ❝♦♥❡①ã♦ ❞❡ ✈♦♦s ❛ér❡♦s✱ ❡♥tr❡ ♦✉tr♦s✳ ❈♦♥t✉❞♦✱ ❡✈✐❞❡♥❝✐❛♠♦s q✉❡ ❡ss❛s ❧✐❣❛çõ❡s ♣♦❞❡♠ s❡r t❛♥❣í✈❡✐s ♦✉ ♥ã♦✱ ❝♦♠♦ ♥♦ ❝❛s♦ ❞❛s r❡❞❡s s❡♠ ✜♦ ❞♦ t✐♣♦ ♠❡s❤
✷ ✲ ❛s ❲▲❆◆✱ ♣r✐♥❝✐♣❛❧♠❡♥t❡ ❞♦ t✐♣♦ ✇✐✲✜✳
❊♠❜♦r❛✱ ❛ t❡♦r✐❛ ❞♦s ❣r❛❢♦s só t❡♥❤❛ s✐❞♦ ❞❡s❡♥✈♦❧✈✐❞❛ ♥♦ sé❝✉❧♦ ♣❛ss❛❞♦✱ ❝♦♠♦ ♠❡♥❝✐♦♥❛♠♦s ❛❝✐♠❛✱ ❡♠ ✶✼✸✻ t✐✈❡♠♦s ❛ ♣✉❜❧✐❝❛çã♦ ❞♦ ♣r♦❜❧❡♠❛ ❞❛s P♦♥t❡s ❞❡ ❑ö♥✐❣s❜❡r❣ ✖ ✉♠ ✐♠♣♦rt❛♥t❡ ♣r♦❜❧❡♠❛ ❞❡ss❛ t❡♦r✐❛✱ q✉❡ s❡rá ❛❜♦r❞❛❞♦ ♥♦ ♣r✐♠❡✐r♦ ❝❛♣ít✉❧♦✳ ❈♦♠ ❛ ♣✉❜❧✐❝❛çã♦ ❞♦s tr❛❜❛❧❤♦s ❞❡ ❋♦r❞ ❡ ❋✉❧❦❡rs♦♥✱ ✶✾✺✻❀ ❇❡r❣❡✱ ✶✾✺✼❀ ❡ ❖r❡✱ ✶✾✻✷✱ ❝r❡s❝❡✉ ♦ ✐♥t❡r❡ss❡ ♥♦ ❡st✉❞♦ ❞♦s ❣r❛❢♦s✱ ❡✱ ❛ ♣❛rt✐r ❞❡ ✶✾✼✵✱ t❡♠♦s ♦ ❞❡s♣♦♥t❛r ❞❛s ♣✉❜❧✐❝❛çõ❡s ❞❡ss❛ t❡♦r✐❛✳ ❚❛♠❜é♠✱ ❛s ❞❡s❝♦❜❡rt❛s ❞❡ ❛♣❧✐❝❛çõ❡s ♣r♦♠♦✈❡r❛♠ ♦ s❡✉ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦✱ ♣♦✐s ❛❧❣♦r✐t♠♦s ❝♦♠♣✉t❛❝✐♦♥❛✐s s❡ t♦r♥❛r❛♠ ♦ ❛❧✈♦ ❞❛s ♣✉❜❧✐❝❛çõ❡s
✶ ❚❡❝♥♦❧♦❣✐❛s ❞❡ ■♥❢♦r♠❛çã♦ ❡ ❈♦♠✉♥✐❝❛çã♦✳
✷ ❘❡❞❡s s❡♠ ✜♦ ❡♠ ♠❛❧❤❛✱ ❡♠❜♦r❛ ❡ss❡ t❡r♠♦ ♥ã♦ s❡❥❛ ♠✉✐t♦ ✉t✐❧✐③❛❞♦ ♥♦ ❇r❛s✐❧✳
❝♦♠ ❛ ❡①♣❛♥sã♦ ❞♦s ❝♦♠♣✉t❛❞♦r❡s✳ ✸ ❖s ♣r✐♠❡✐r♦s r❡s✉❧t❛❞♦s s♦❜r❡ ❣r❛❢♦s ✈✐❡r❛♠ ❞❛
❊✉r♦♣❛✱ r❡❣✐ã♦ q✉❡ s✉r❣✐✉ ❛ ♣r✐♠❡✐r❛ ❡s❝♦❧❛✿ ❆ ❤ú♥❣❛r❛ ✭❝♦♠ ❛ ❝♦♥tr✐❜✉✐çã♦ ❞❡ ✈ár✐♦s ♣❡sq✉✐s❛❞♦r❡s✱ ❡♥tr❡ ❡❧❡s✱ ❊r❞♦s✱ ❍❛❥♥á❧✱ ❚✉rá♥✮✳ ▲♦❣♦ ❛♣ós ♥❛s❝❡♠ ❛ ❡s❝♦❧❛ ❢r❛♥❝❡s❛✱ ❛ ❛♠❡r✐❝❛♥❛✳ ❖✉tr♦s ♥♦♠❡s ✐♠♣♦rt❛♥t❡s ❛✢♦r❛♠ ♣❛r❛ ♦ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ❞❡st❛ t❡♦r✐❛ ♥❛ ❡①✲❯♥✐ã♦ ❙♦✈✐ét✐❝❛✱ ❈❛♥❛❞á ❡ ■♥❣❧❛t❡rr❛✳ ❈♦♠ ❛ ❝r✐❛çã♦ ❞❛s ❡s❝♦❧❛s ✐♥❞❡♣❡♥❞❡♥t❡s ❡ ♦ ❣r❛♥❞❡ ♥ú♠❡r♦ ❞❡ tr❛❜❛❧❤♦s ♣✉❜❧✐❝❛❞♦s ❡♠ ❝✉rt♦ ❡s♣❛ç♦ ❞❡ t❡♠♣♦✱ s✉r❣❡ ✉♠❛ ❣r❛♥❞❡ ✈❛r✐❡❞❛❞❡s ❞❡ ♥♦♠❡♥❝❧❛t✉r❛s✱ ❞✐✜❝✉❧t❛♥❞♦ ♠✉✐t❛s ✈❡③❡s ❛ ❝♦♠♣r❡❡♥sã♦✳ ❉✐✈❡rs✐❞❛❞❡ ❡ss❛ q✉❡✱ ❛ ♣r✐♥❝í♣✐♦✱ ❞✐✜❝✉❧t♦✉ ❛ ❝♦♥str✉çã♦ ❞♦ tr❛❜❛❧❤♦ ♦r✐✉♥❞♦ ❞♦ ❡st✉❞♦ ❛q✉✐ tr❛t❛❞♦✳ P❛r❛ ✐❧✉str❛r ♦ q✉❡ ❛❝❛❜❛♠♦s ❞❡ ❞✐③❡r✱ ♣♦❞❡rã♦ s❡r ❡♥❝♦♥tr❛❞❛s ♥❛ ❧✐t❡r❛t✉r❛ ❛❢❡✐t❛ ❞❡✜♥✐çõ❡s ❞✐st✐♥t❛s ❞❡ ♣❛ss❡✐♦s✱ ♠✉❧t✐❣r❛❢♦s✱ ❝❛♠✐♥❤♦s ❡ ♦✉tr♦s ❝♦♥❝❡✐t♦s ❝♦rr❡❧❛t♦s✳
■st♦ ♣♦st♦✱ ♦ ♣r✐♠❡✐r♦ ❝❛♣ít✉❧♦ tr❛t❛rá ❞❛s ❞❡✜♥✐çõ❡s ❡✱ ❝♦♥❝♦♠✐t❛♥t❡♠❡♥t❡✱ tr❛r❡♠♦s ❛❧❣✉♥s ❡①❡♠♣❧♦s ❞♦ ❧✐✈r♦ ❈ír❝✉❧♦s ▼❛t❡♠át✐❝♦s ✭❬✺❪✮✱ ❛ ✜♠ ❞❡ ♠♦str❛r ❝♦♠♦ ❛ t❡♦r✐❛ ❞♦s ❣r❛❢♦s é út✐❧ ♥❛ r❡s♦❧✉çã♦ ❞❡ ♣r♦❜❧❡♠❛s✱ ♦s q✉❛✐s ♠✉✐t❛s ✈❡③❡s ♥❡♠ ♣❛r❡❝❡♠ t❡r r❡❧❛çã♦ ❝♦♠ ❛ ♠❛tér✐❛✳
❖✉tr♦ ✐♠♣♦rt❛♥t❡ ❞❡st❛q✉❡ ❞❡st❡ ❝❛♠♣♦ ♠❛t❡♠át✐❝♦ é ❛ ❝♦❧♦r❛çã♦ ❞❡ ✈ért✐❝❡s ❡ ❞❡ ❛r❡st❛s✱ q✉❡ s❡rã♦ ❛❜♦r❞❛❞♦s ♥♦ s❡❣✉♥❞♦ ❝❛♣ít✉❧♦✳ ❚r❛t❛r❡♠♦s ❞♦ t❡♦r❡♠❛ ❞❛s q✉❛tr♦ ❝♦r❡s✱ ❢❛③❡♥❞♦ ✉♠ ♣❡q✉❡♥♦ r❡❧❛t♦ ❤✐stór✐❝♦ ❡ ❛♣r❡s❡♥t❛♥❞♦ ❛♣❧✐❝❛çõ❡s✳ ◗✉❛♥t♦ ❛ ❝♦❧♦r❛çã♦ ❞❡ ❛r❡st❛s✱ ✈❡rs❛r❡♠♦s s♦❜r❡ ♦ ♥ú♠❡r♦ ❞❡ ❘❛♠s❡②✱ q✉❡ t♦r♥♦✉✲s❡ ❞❡ ❡①tr❡♠❛ r❡❧❡✈â♥❝✐❛ ♥❛ ❝♦♠❜✐♥❛tór✐❛✱ ❛ ♣♦♥t♦ ❞❡ t♦r♥❛r✲s❡ ✉♠❛ s✉❜❞✐s❝✐♣❧✐♥❛✳
P♦r ú❧t✐♠♦✱ ♥♦ t❡r❝❡✐r♦ ❝❛♣ít✉❧♦ ❢❛r❡♠♦s ✉♠ ❡st✉❞♦ ❞♦ ♠ét♦❞♦ ♣r♦❜❛❜✐❧íst✐❝♦✱ ❛♣❧✐❝❛♥❞♦ ❛ ♣r♦❜❛❜✐❧✐❞❛❞❡ à t❡♦r✐❛ ❞♦s ❣r❛❢♦s✱ q✉❡ ♣♦ss✉✐ ❡❧❡✈❛❞❛ ✉t✐❧✐❞❛❞❡✱ ♣♦✐s ♥❡♠ s❡♠♣r❡ ❝♦♥s❡❣✉✐♠♦s ❝r✐❛r ✉♠ ❣r❛❢♦ ❝♦♠ ❝❡rt❛s ♣r♦♣r✐❡❞❛❞❡s✱ s❡♥❞♦ ♥❡❝❡ssár✐♦ ❧❛♥ç❛r♠♦s ♠ã♦ ❞❡ t❡♦r❡♠❛s q✉❡ ✈❛❧✐❞❛♠ ❛ s✉❛ ❡①✐stê♥❝✐❛✱ ♦✉ ❛✐♥❞❛ ❝♦♠♣r♦✈❛r q✉❡ ❛ ♣r♦❜❛❜✐❧✐❞❛❞❡ ❞❡ s✉❛ ♦❝♦rrê♥❝✐❛ é ♠❛✐♦r q✉❡ ③❡r♦✳ ◆❡st❡ ❝♦♥t❡①t♦✱ r❡t♦r♥❛r❡♠♦s ❛♦ ♥ú♠❡r♦ ❞❡ ❘❛♠s❡②✳ ❙❡❣✉♥❞❛s ✈❡③❡s✱ ♣r♦✈❛❞❛ ❛ ❡①✐stê♥❝✐❛ ❞❡ ✉♠ ❣r❛❢♦ ❝♦♠ ❞❡t❡r♠✐♥❛❞❛s ❝❛r❛❝t❡ríst✐❝❛s✱ ♣♦❞❡♠♦s ❝♦♥str✉✐r ♦ ❣r❛❢♦ q✉❡ ♣r❡❝✐s❛♠♦s✳ ❉❡s❡♥✈♦❧✈❡r❡♠♦s ❡st❡ ❡st✉❞♦ ❛tr❛✈és ❞♦ t❡①t♦ ✧Pr♦❜❛❜✐❧✐❞❛❞❡ ❛♣❧✐❝❛❞❛ à t❡♦r✐❛ ❞♦s ❣r❛❢♦s ✭✈❡r r❡❢❡rê♥❝✐❛ ❬✶✼❪✮✱ tr❛❜❛❧❤❛♥❞♦ ♦s ❡①❡♠♣❧♦s ❛♣r❡s❡♥t❛❞♦s✱ ✜♥❛❧✐③❛♥❞♦ ❝♦♠ ❛ ❞❡♠♦♥str❛çã♦ ❞❡ ✉♠ ❞♦s ❚❡♦r❡♠❛s ❞❡ P❛✉❧
✸■♥❢♦r♠❛çõ❡s r❡t✐r❛❞❛s ❞❛ r❡❢❡rê♥❝✐❛ ❬✷❪
❊r❞♦s✳
➚ ✈✐st❛ ❞✐ss♦✱ ✜❝❛♥❞♦ ❞✐♠❡♥s✐♦♥❛❞❛ ❛ ♠❛❣♥✐t✉❞❡ ❞❛ t❡♠át✐❝❛ ❡♠ ❝♦♠❡♥t♦✱ t❡♠♦s ❝♦♠♦ ♦❜❥❡t✐✈♦ ❡str❡✐t❛r✱ t♦r♥❛r ♠❛✐s ❛❧❝❛♥çá✈❡❧ ❡ ❝♦♥❤❡❝✐❞❛✱ ❛♦s ♣❛rtí❝✐♣❡s ❞❛ ❡❞✉❝❛çã♦ ❜ás✐❝❛ ✭❡❞✉❝❛♥❞♦ ❡ ♣r♦❢❡ss♦r✮ ❛s ❞❡✜♥✐çõ❡s✱ ❝♦♥❝❡✐t♦s ❡ ✐❞❡✐❛s s♦❜r❡ ❣r❛❢♦s✱ ♣❛r❛ q✉❡ ❡ss❡s ♣♦ss❛♠ ❢♦♠❡♥t❛r ❡st✉❞♦s ❢✉t✉r♦s ❡ ♣r♦✈❡✐t♦s♦s à ❝♦♠✉♥✐❞❛❞❡ ❛❜r❛♥❣❡♥t❡✱ ♠❛s ♥ã♦ s♦♠❡♥t❡✳ ❖✉tr♦ss✐♠✱ ✉♠❛ ✈❡③ q✉❡ ❛ t❡♦r✐❛ ❞♦s ❣r❛❢♦s é ❞❡ ❢á❝✐❧ ❝♦♠♣r❡❡♥sã♦✱ t❡♠♦s ❛✐♥❞❛ ❝♦♠♦ ♠❡t❛ ❞❡s❡♥✈♦❧✈❡r ❡str✉t✉r❛s ❝♦❣♥✐t✐✈❛s ❢❛✈♦rá✈❡✐s à r❡s♦❧✉çã♦ ❞❡ ✉♠❛ ♠✉❧t✐♣❧✐❝✐❞❛❞❡ ❞❡ ♣r♦❜❧❡♠❛s ♠❛t❡♠át✐❝♦s✱ ❝♦♠ ❜❛s❡ ♥❛ ❡①♣❧♦r❛çã♦ ❞❡st❡ s❛❜❡r✱ ✐st♦ é✱ ✈✐s❛♠♦s t♦r♥❛r ❡st❛ ♣rát✐❝❛ ✉♠❛ ❢❡rr❛♠❡♥t❛ ❛♣r✐♠♦r❛❞♦r❛ ❞♦ ♣❡♥s❛♠❡♥t♦ ♠❛t❡♠át✐❝♦✳ P♦r ✜♠✱ ♠✐r❛♠♦s ♣r♦♠♦✈❡r ❝♦♥❞✐çõ❡s ♣❛r❛ ♦ ❝♦♥t❛t♦ ✐♥✐❝✐❛❧ ❞♦s ❞✐s❝❡♥t❡s ❝♦♠ ❛ ♣❡sq✉✐s❛ ❝✐❡♥tí✜❝❛ ❡♠ ❝✐r❝✉♥stâ♥❝✐❛s ❛♥t❡r✐♦r❡s ❛♦ ❡♥s✐♥♦ s✉♣❡r✐♦r✱ ❞❛❞❛ ❛ s✉❛ ✐♠♣♦rtâ♥❝✐❛ ♥♦ ♣r♦❝❡ss♦ ❞❡ ❡♥s✐♥♦ ❡ ❛♣r❡♥❞✐③❛❣❡♠✱ ✉♠❛ ✈❡③ q✉❡ ❛ t❡♦r✐❛ ❞♦s ❣r❛❢♦s é ❢r✉t✉♦s❛ à ♣❡sq✉✐s❛ ❝✐❡♥tí✜❝❛✱ ♥♦s ❛♣r♦①✐♠❛♥❞♦ ❡①♣r❡ss✐✈❛♠❡♥t❡ ❞❛ ♠❛t❡♠át✐❝❛ ♠❛✐s ❛❜str❛t❛ q✉❡ é ✈✐st❛ ♥♦ ❡♥s✐♥♦ s✉♣❡r✐♦r✳
❈❛♣ít✉❧♦ ✶
❚❊❖❘■❆ ❉❖❙ ●❘❆❋❖❙
◆❡st❡ ❝❛♣ít✉❧♦✱ ❛❜♦r❞❛r❡♠♦s ❛ t❡♦r✐❛ ❞♦s ❣r❛❢♦s✱ ❛s ❞❡✜♥✐çõ❡s ❡ ❛r❣✉♠❡♥t♦s ❡❧❡♠❡♥t❛r❡s ❢❛③❡♥❞♦ ✉♠ ❛♣❛♥❤❛❞♦ ❞♦s r❡s✉❧t❛❞♦s ❛♣r❡s❡♥t❛❞♦s ♥♦s ❝❛♣ít✉❧♦s ✺ ❡ ✶✸ ❞♦ ❧✐✈r♦ ❈ír❝✉❧♦s ▼❛t❡♠át✐❝♦s✱ ✭r❡❢❡rê♥❝✐❛ ❬✺❪✮✱ ❝♦♠ ❡①♣❧✐❝❛çõ❡s ❡ ❝♦♠❡♥tár✐♦s✱ r❡s♦❧✈❡♥❞♦ ❛❧❣✉♥s ❡①❡r❝í❝✐♦s q✉❡ sã♦ ❛♣r❡s❡♥t❛❞♦s✳
❉❡✜♥✐çã♦ ✶ ❯♠ ❣r❛❢♦ ●✭❱✱❊✮ é ❝♦♠♣♦st♦ ❞❡ ✉♠ ❝♦♥❥✉♥t♦ ✜♥✐t♦ ❡ ♥ã♦ ✈❛③✐♦ ❞❡ ✈ért✐❝❡s ✭❱✮✱ ❡ ✉♠ ❝♦♥❥✉♥t♦ ❞❡ ♣❛r❡s ♥ã♦ ♦r❞❡♥❛❞♦s ❞❡st❡s ✈ért✐❝❡s ✭❊✮✲ ❝❤❛♠❛❞♦s ❞❡ ❛r❡st❛s✳
❊♠ ♦✉tr❛s ♣❛❧❛✈r❛s✱ ✉♠ ❣r❛❢♦ ❝♦♥s✐st❡ ♥❛ r❡♣r❡s❡♥t❛çã♦ ❞❡ ✉♠ ❝♦♥❥✉♥t♦✱ ♥♦ q✉❛❧ ❝❛❞❛ ✉♠ ❞♦s s❡✉s ❡❧❡♠❡♥t♦s s❡ r❡❧❛❝✐♦♥❛ ♦✉ ♥ã♦ ❝♦♠ ♦s ♦✉tr♦s ❡❧❡♠❡♥t♦s ❞❡ss❡ ♠❡s♠♦ ❝♦♥❥✉♥t♦✳ ❖s ❡❧❡♠❡♥t♦s ✭♦❜❥❡t♦s✮ ❝♦♥t✐❞♦s ♥❡ss❡ ❝♦♥❥✉♥t♦ sã♦ ❝❤❛♠❛❞♦s ❞❡ ✈ért✐❝❡s ❞♦ ❣r❛❢♦❀ ❛s ❧✐❣❛çõ❡s ✭r❡❧❛❝✐♦♥❛♠❡♥t♦s✮ ❡♥tr❡ ♦s ❡❧❡♠❡♥t♦s sã♦ ❝❤❛♠❛❞♦s ❞❡ ❛r❡st❛s ❞♦s ❣r❛❢♦s✳
❉❡✜♥✐çã♦ ✷ ❯♠ ❣r❛❢♦ ❢♦r♠❛❞♦ ❛♣❡♥❛s ♣♦r ✉♠ ✈ért✐❝❡ r❡❝❡❜❡ ♦ ♥♦♠❡ ❞❡ ❣r❛❢♦ tr✐✈✐❛❧❀ ✈❡❥❛ ✜❣✉r❛ ✶✳✶ ✭❛✮✳
❉❡✜♥✐çã♦ ✸ ❯♠ ❣r❛❢♦ é ❞✐t♦ ❝♦♠♣❧❡t♦ q✉❛♥❞♦ t♦❞♦ ♣❛r ❞❡ ✈ért✐❝❡s é ❧✐❣❛❞♦ ♣♦r ✉♠❛ ❛r❡st❛✳ ❊ ♦ ❞❡♥♦t❛♠♦s ♣♦r Kn✳ ◆❛ ✜❣✉r❛ ✶✳✶ ✭❜✮ t❡♠♦s ✉♠ ❣r❛❢♦ k5✳
❈❛♣ít✉❧♦ ✶✳ ❚❊❖❘■❆ ❉❖❙ ●❘❆❋❖❙
❋✐❣✉r❛ ✶✳✶✿ ❊①❡♠♣❧♦s ❞❡ ●r❛❢♦s✿ ✭❛✮ ❣r❛❢♦ tr✐✈✐❛❧✱ ✭❜✮ ❣r❛❢♦ ❝♦♠♣❧❡t♦ ❡ ✭❝✮ ♣s❡✉❞♦❣r❛❢♦✳
✭❛✮ ✭❜✮ ✭❝✮
◗✉❛♥❞♦ ❞♦✐s ✈ért✐❝❡s sã♦ ❧✐❣❛❞♦s ♣♦r ✉♠❛ ❛r❡st❛ ❞✐③❡♠♦s q✉❡ ❡❧❡s sã♦ ❛❞❥❛❝❡♥t❡s ❡ q✉❡ ❡st❛ ❛r❡st❛ é ✐♥❝✐❞❡♥t❡ ❛♦s ✈ért✐❝❡s✳ ❯♠ ❛r❝♦ ❛ss♦❝✐❛❞♦ ❛♦ ♠❡s♠♦ ✈ért✐❝❡ é ❝❤❛♠❛❞♦ ❞❡ ✧❧♦♦♣✧✱ q✉❡ ❡♠ ♣♦rt✉❣✉ês s✐❣♥✐✜❝❛ ❧❛ç♦✳ ❉❡ ❛❝♦r❞♦ ❝♦♠ ❍❛r❛r②✶✱ q✉❛♥❞♦ ✉♠ ❣r❛❢♦
♣♦ss✉✐ ♠❛✐s ❞❡ ✉♠❛ ❛r❡st❛ ❧✐❣❛♥❞♦ ♦ ♠❡s♠♦ ♣❛r ❞❡ ✈ért✐❝❡✱ s❡♠ ❧♦♦♣✱ ❡❧❡ é ❞❡♥♦♠✐♥❛❞♦ ♠✉❧t✐❣r❛❢♦✳ ❊ s❡✱ ❛❧é♠ ❞❡ ♠ú❧t✐♣❧❛s✱ ❡❧❡ ♣♦ss✉✐r ❧♦♦♣✱ r❡❝❡❜❡ ♦ ♥♦♠❡ ❞❡ ♣s❡✉❞♦❣r❛❢♦✱ ❝♦♥❢♦r♠❡ ❡①❡♠♣❧♦ ❛♥t❡r✐♦r ✐❧✉str❛❞♦ ♥❛ ✜❣✉r❛ ✶✳✶ ✭❝✮❀ ♥❛ q✉❛❧ t❡♠♦s ✉♠ ❧♦♦♣ ♥♦ ✈ért✐❝❡ ❋ ❡ ❛r❡st❛s ♠ú❧t✐♣❧❛s ✉♥✐♥❞♦ ♦s ✈ért✐❝❡s ● ❡ ❍✳ ❯♠ ❣r❛❢♦ s❡♠ ❧❛ç♦s ❡ s❡♠ ❛r❡st❛s ♠ú❧t✐♣❧❛s é ❝❤❛♠❛❞♦ ❞❡ ❣r❛❢♦ s✐♠♣❧❡s✳
❊①❡♠♣❧♦ ✶ ❯♠ t❛❜✉❧❡✐r♦ ❝♦♠ ❛ ❢♦r♠❛ ❞❡ ✉♠❛ ❝r✉③ é ♦❜t✐❞♦ ❞❡ ✉♠ t❛❜✉❧❡✐r♦ 4×4
r❡t✐r❛♥❞♦✲s❡ ❛s q✉❛tr♦ q✉✐♥❛s✳ ❯♠ ❝❛✈❛❧♦ ♣♦❞❡ s❡ ♠♦✈❡r ✷ ♥❡st❡ t❛❜✉❧❡✐r♦ ❞❡ ♠♦❞♦ ❛
♣❛r❛r ❡♠ t♦❞♦s ♦s q✉❛❞r❛❞♦s ❡①❛t❛♠❡♥t❡ ✉♠❛ ✈❡③ ❡ t❡r♠✐♥❛r ♥♦ q✉❛❞r❛❞♦ ❞❡ ♦♥❞❡ s❛✐✉❄
P❛r❛ ✉♠❛ ♠❡❧❤♦r ❝♦♠♣r❡❡♥sã♦✱ ♥♦♠❡❡♠♦s ♦s q✉❛❞r✐♥❤♦s ❞♦ t❛❜✉❧❡✐r♦ ❞❛ ❢♦r♠❛ ❛❜❛✐①♦✿
✶◆♦♠❡♥❝❧❛t✉r❛ ❝♦♠ ❜❛s❡ ♥❛ ❘❡❢❡rê♥❝✐❛ ❬✶✸❪✳ ❙❡❣✉♥❞♦ ♦✉tr❛s ❧✐t❡r❛t✉r❛s✱ ♦ ❣r❛❢♦ r❡❝❡❜❡ ♦ ♥♦♠❡ ❞❡
♠✉❧t✐❣r❛❢♦s ♠❡s♠♦ ❝♦♥t❡♥❞♦ ❧♦♦♣s✳
✷❖s ♠♦✈✐♠❡♥t♦s ❞♦ ❝❛✈❛❧♦ ❝♦♥s✐st❡♠ ❡♠ ♠♦✈❡r✲s❡ ✉♠❛ ú♥✐❝❛ ❝❛s❛ ♥❛ ❤♦r✐③♦♥t❛❧✱ ❡ ❡♠ s❡❣✉✐❞❛ ❞✉❛s
❝❛s❛s ♥❛ ✈❡rt✐❝❛❧✱ ♦✉ ✉♠❛ ❝❛s❛ ♥❛ ✈❡rt✐❝❛❧ ❡ ♣♦st❡r✐♦r♠❡♥t❡ ❞✉❛s ❝❛s❛s ♥❛ ❤♦r✐③♦♥t❛❧✳
❈❛♣ít✉❧♦ ✶✳ ❚❊❖❘■❆ ❉❖❙ ●❘❆❋❖❙
❖❜s❡r✈❡♠♦s q✉❡ s❡ ✉t✐❧✐③❛r♠♦s ❛ ♠♦❞❡❧❛❣❡♠ ❝♦♠ ❣r❛❢♦s✱ ♦ ♣r♦❜❧❡♠❛ ✜❝❛ ❜❡♠ ♠❛✐s s✐♠♣❧❡s✳ ❚❡♠♦s ✶✷ ✈ért✐❝❡s✱ r❡st❛ ❞❡s❝♦❜r✐r q✉❛✐s ❞❡❧❡s ❡stã♦ ❧✐❣❛❞♦s ♣♦r ❛r❡st❛s✳ ❖ ✈ért✐❝❡ ❆ s❡ ❧✐❣❛ ❛♦ ■✱ ❛♦ ●✱ ❡ ❛♦ ❋❀ ♦ ■ ♣♦r s✉❛ ✈❡③ ❛♣❡♥❛s ❛♦ ❈ ♦✉ ❛♦ ❆❀ ♦ ❈ ♣♦r s✉❛ ✈❡③ s❡ ❧✐❣❛ ❛♦ ❑ ❡ ❛♦ ♣ró♣r✐♦ ■✱ ♦ ❑ s❡ ❧✐❣❛ ❛♦ ❏ ❡ ❛♦ ▲❀ ♦ ❏ ❛♦ ❉ ❡ ❛♦ ❇❀ ♦ ❋ s❡ ❧✐❣❛ ❛♦ ▲ ❡ ❛♦ ❍❀ ❡✱ ❛ss✐♠ ♣♦r ❞✐❛♥t❡✱ ♣♦❞❡♠♦s ✈❡r✐✜❝❛r ❛ r❡❧❛çã♦ ❞❡ ❝❛❞❛ ✉♠ ❞♦s ✈ért✐❝❡s ❝♦♠ ♦s ♦✉tr♦s✱ ♦ q✉❡ r❡s✉❧t❛ ♥♦ ❣r❛❢♦ ❞❛ ✜❣✉r❛ ✶✳✷✳ P❛r❛ ✐ss♦✱ ❜❛st❛ ❡s❝♦❧❤❡r♠♦s ✉♠ ❞♦s ✈ért✐❝❡s ♣❛r❛ ❝♦♠❡ç❛r ❡ s❡❣✉✐r ♣❡❧♦s ♦✉tr♦s ❡♠ s❡♥t✐❞♦ ❤♦rár✐♦✱ ♣♦✐s✱ ❞❡st❛ ❢♦r♠❛✱ ✜❝❛rá ❝❧❛r♦ q✉❡ ♦ ❝❛✈❛❧♦ ♣♦❞❡ s❡ ♠♦✈❡r ♣❛ss❛♥❞♦ ♣♦r t♦❞♦s ♦s q✉❛❞r❛❞♦s ❡①❛t❛♠❡♥t❡ ✉♠❛ ✈❡③ ❡ t❡r♠✐♥❛r ♥♦ q✉❛❞r❛❞♦ ❞❡ ♦♥❞❡ ✐♥✐❝✐♦✉✳
❋✐❣✉r❛ ✶✳✷✿ ●r❛❢♦ ❞♦ t❛❜✉❧❡✐r♦
❊①❡♠♣❧♦ ✷ ❋✐❣✉r❛t✐✈♦ é ✉♠ ♣❛ís ❝♦♠ ♥♦✈❡ ❝✐❞❛❞❡s ❞❡ ♥♦♠❡s ✶✱ ✷✱ ✸✱ ✹✱ ✺✱ ✻✱ ✼✱ ✽✱ ✾✳ ❯♠ ✈✐❛❥❛♥t❡ ❞❡s❝♦❜r❡ q✉❡ ❡①✐st❡ ✈♦♦ ❞✐r❡t♦ ❞❡ ✉♠❛ ❝✐❞❛❞❡ ❛ ♦✉tr❛ s❡ ❡ s♦♠❡♥t❡ s❡ ♦ ♥ú♠❡r♦ ❞❡ ❞♦✐s ❛❧❣❛r✐s♠♦s ❢♦r♠❛❞♦s ♣❡❧♦s ♥♦♠❡s ❞❛s ❝✐❞❛❞❡s é ❞✐✈✐sí✈❡❧ ♣♦r ✸✳ ❖ ✈✐❛❥❛♥t❡ ♣♦❞❡ ✐r ❞❛ ❈✐❞❛❞❡ ✶ ♣❛r❛ ❛ ❈✐❞❛❞❡ ✾❄
❈❛♣ít✉❧♦ ✶✳ ❚❊❖❘■❆ ❉❖❙ ●❘❆❋❖❙ ✶✳✶✳ ●r❛✉ ❞♦ ✈ért✐❝❡
▲❡♠❜r❡♠♦s q✉❡ ♣❛r❛ q✉❡ ✉♠ ♥ú♠❡r♦ s❡❥❛ ❞✐✈✐sí✈❡❧ ♣♦r ✸✱ ❛ s♦♠❛ ❞❡ s❡✉s ❛❧❣❛r✐s♠♦s t❡♠ q✉❡ s❡r ✉♠ ♥ú♠❡r♦ ❞✐✈✐sí✈❡❧ ♣♦r ✸✳ ▲♦❣♦✱ ❛ ❝✐❞❛❞❡ ✶ ❡stá ❧✐❣❛❞❛ ❛ ❈✐❞❛❞❡ ✷✱ ✺ ❡ ✽✳ ❊ ❛ ❝✐❞❛❞❡ ✾ só ❡stá ❧✐❣❛❞❛ ❛ ❝✐❞❛❞❡ ✸ ❡ ❛ ❝✐❞❛❞❡ ✻✳ ❊ ❛ ✸✱ ❛♣❡♥❛s ❛ ✻ ❡ ❛ ✾❀ ❡ ❛ ✻✱ ❛♣❡♥❛s ❛ ✸ ❡ ❛ ✾✳ P♦rt❛♥t♦ ♥ã♦ ❤á ❝♦♠♦ ✐r ❞❛ ❝✐❞❛❞❡ ✶ ❛ ✾✳ ❖ ❣r❛❢♦ ❞❛ ✜❣✉r❛ ❛❜❛✐①♦ r❡♣r❡s❡♥t❛ ❜❡♠ ❡st❡ ♣r♦❜❧❡♠❛✱ ♠♦str❛♥❞♦ q✉❡ ♥ã♦ é ♣♦ssí✈❡❧ ❛♦ ✈✐❛❥❛♥t❡ ✐r ❞❛ ❝✐❞❛❞❡ ✶ à ❝✐❞❛❞❡ ✾✳
❋✐❣✉r❛ ✶✳✸✿ ●r❛❢♦ ♣r♦❜❧❡♠❛ ✷
❖ ❣r❛❢♦ ❞❡ss❡ ♣r♦❜❧❡♠❛ é ❞❡s❝♦♥❡①♦✱ ♣♦✐s ♥ã♦ ❡①✐st❡ ✉♠ ❝❛♠✐♥❤♦ ❞♦ ✈ért✐❝❡ ✧✾✧ ♣❛r❛ ♦ ✧✶✧ ♣♦r ❡①❡♠♣❧♦✳ ▼❛s✱ só ❞❡✜♥✐r❡♠♦s ❡ss❡s ❝♦♥❝❡✐t♦s ♥❛ s❡çã♦ ✶✳✷✳
✶✳✶ ●r❛✉ ❞♦ ✈ért✐❝❡
❉❡✜♥✐çã♦ ✹ ❖ ❣r❛✉ ❞❡ ✉♠ ✈ért✐❝❡ ✈ é ❞❡t❡r♠✐♥❛❞♦ ♣❡❧♦ ♥ú♠❡r♦ ❞❡ ❛r❡st❛s q✉❡ ✐♥❝✐❞❡♠ ♥❡❧❡✱ ♦✉ s❡❥❛✱ é ♦ ♥ú♠❡r♦ ❞❡ ✈❡rtí❝❡s ❛❞❥❛❝❡♥t❡s ❛ ✈✱ ❡ é ❞❡♥♦t❛❞♦ ♣♦r dg(v)✳
◆❛ ✜❣✉r❛ ✶✳✹ ♦ ✈ért✐❝❡ ❈ t❡♠ ❣r❛✉ ✹✱ ♦ ❋ t❡♠ ❣r❛✉ ✵✱ ❇ ❡ ❉ t❡♠ ❣r❛✉ ✷✱ ❡ ♦s ♦✉tr♦s t❡♠ ❣r❛✉ ✸✳
◗✉❛♥❞♦ t♦❞♦s ♦s ✈ért✐❝❡s ❞❡ ✉♠ ❣r❛❢♦ tê♠ ✉♠ ♠❡s♠♦ ❣r❛✉ r✱ ❡❧❡ r❡❝❡❜❡ ♦ ♥♦♠❡ ❞❡ r✲r❡❣✉❧❛r✳ ❖ ●r❛❢♦ ❍ ❞❛ ✜❣✉r❛ ✶✳✹ é ✸✲r❡❣✉❧❛r✳
❖ ♠❡♥♦r ❣r❛✉ ♣r❡s❡♥t❡ ❡♠ ✉♠ ❣r❛❢♦ ● é ❞❡♥♦t❛❞♦ ♣♦rδ(G)❀ ❖ ♠❛✐♦r ❣r❛✉ ❡①✐st❡♥t❡
❡♠ ✉♠ ❣r❛❢♦ ● é ❞❡♥♦t❛❞♦ ♣♦r ∆(G)✳ ◆❛ ✜❣✉r❛ ✶✳✹δ(G) = 0 ❡∆(G) = 4
❯♠ ✈ért✐❝❡ é ❞✐t♦ ✈ért✐❝❡ ♣❛r s❡ ♦ s❡✉ ❣r❛✉ ❢♦r ♣❛r✱ ❡ ✈ért✐❝❡ í♠♣❛r✱ s❡ s❡✉ ❣r❛✉ ❢♦r í♠♣❛r✳
❈❛♣ít✉❧♦ ✶✳ ❚❊❖❘■❆ ❉❖❙ ●❘❆❋❖❙ ✶✳✶✳ ●r❛✉ ❞♦ ✈ért✐❝❡
❋✐❣✉r❛ ✶✳✹✿ ❊①❡♠♣❧✐✜❝❛♥❞♦ ❣r❛✉ ❞♦s ✈ért✐❝❡s ❡ ❣r❛❢♦ r✲r❡❣✉❧❛r
❊①❡♠♣❧♦ ✸ ❯♠ ❞❡t❡r♠✐♥❛❞♦ r❡✐♥❛❞♦ t❡♠ ✶✵✵ ❝✐❞❛❞❡s ❡ s❛❡♠ q✉❛tr♦ ❡str❛❞❛s ❞❡ ❝❛❞❛ ✉♠❛ ❞❡❧❛s✳ ◗✉❛♥t❛s ❡str❛❞❛s ❡①✐st❡♠ ❛♦ t♦❞♦ ♥❡st❡ r❡✐♥❛❞♦❄
❊♠ ♥♦ss❛ ♠♦❞❡❧❛❣❡♠✱ ❛s ❝✐❞❛❞❡s sã♦ ♦s ✈ért✐❝❡s ❞♦ ❣r❛❢♦✱ ❡ ❛s ❡str❛❞❛s✱ ❛s ❛r❡st❛s✳ ❖ ♥ú♠❡r♦ ❞❡ ❡str❛❞❛s q✉❡ s❛❡♠ ❞❡ ❝❛❞❛ ❝✐❞❛❞❡ é ♦ ❣r❛✉ ❞♦ ✈ért✐❝❡✳ ❙♦♠❛♥❞♦ ♦ ❣r❛✉ ❞❡ ❝❛❞❛ ✈ért✐❝❡✱ ♦❜t❡♠♦s ✹✵✵✳ P♦ré♠✱ ❝♦♠♦ ❝❛❞❛ ❛r❡st❛ ❧✐❣❛ ✷ ✈ért✐❝❡s ✭❝✐❞❛❞❡s✮✱ ♣r❡❝✐s❛♠♦s ❞✐✈✐❞✐r ♣♦r ✷✳ ▲♦❣♦✱❞ ❡ss❡ r❡✐♥❛❞♦ ♣♦ss✉✐ ✷✵✵ ❡str❛❞❛s✳
❚❡♦r❡♠❛ ✶ ❊♠ ✉♠ ❣r❛❢♦ ●✱ ❛ s♦♠❛ ❞♦s ❣r❛✉s ❞❡ t♦❞♦s ♦s ✈ért✐❝❡s é ✐❣✉❛❧ ❛ ✷ ✈❡③❡s ♦ ♥ú♠❡r♦ ❞❡ ❛r❡st❛s✳
❉❡♠♦♥str❛çã♦✿ ❆ ❞❡♠♦♥str❛çã♦ é ✐♠❡❞✐❛t❛✱ ❝♦♠♦ ❢♦✐ ✈✐st♦ ♥♦ ♣r♦❜❧❡♠❛ ❛♥t❡r✐♦r✿ ❆♦ s♦♠❛r t♦❞♦s ♦s ❣r❛✉s ❞♦s ✈ért✐❝❡s✱ ❡st❛♠♦s s♦♠❛♥❞♦ t♦❞❛s ❛s ❛r❡st❛s ✐♥❝✐❞❡♥t❡s ❛♦s ✈ért✐❝❡s✱ ♠❛s ❝♦♠♦ ❝❛❞❛ ❛r❡st❛ ✐♥❝✐❞❡ ❡♠ ✷ ✈ért✐❝❡s✱ ❡st❛s sã♦ ❝♦♥t❛❞❛s ✷ ✈❡③❡s✱ ❛♦ ❞✐✈✐❞✐r♠♦s ♣♦r ✷ t❡♠♦s ♦ ♥ú♠❡r♦ ❞❡ ❛r❡st❛s✳
❊①❡♠♣❧♦ ✹ ➱ ♣♦ssí✈❡❧ ❡①✐st✐r❡♠ ❡①❛t❛♠❡♥t❡ ✶✵✵ ❡str❛❞❛s ❡♠ ✉♠ r❡✐♥❛❞♦ ♥♦ q✉❛❧ ❡①✐st❡♠ ✸ ❡str❛❞❛s s❛✐♥❞♦ ❞❡ ❝❛❞❛ ❝✐❞❛❞❡❄
P❡♥s❡♠♦s ❡♠ ✉♠ ❣r❛❢♦ ❝♦♠ ✶✵✵ ❛r❡st❛s (E)✱ ♥ã♦ s❛❜❡♠♦s ♦ ♥ú♠❡r♦ ❞❡ ✈ért✐❝❡s (V)✱ ♠❛s ❝❛❞❛ ✉♠ ❞♦s ✈ért✐❝❡s t❡♠ ❣r❛✉ ✸✳ ▲♦❣♦✱ ♣❡❧♦ t❡♦r❡♠❛ ✶✿
2·E = 3·V ⇒200 = 3·V,
❞♦♥❞❡ ❞❡❝♦rr❡ q✉❡ ❱ ♥ã♦ é ✉♠ ♥ú♠❡r♦ ✐♥t❡✐r♦✳ P♦rt❛♥t♦✱ ♥ã♦ é ♣♦ssí✈❡❧ ❡①✐st✐r❡♠ ❡①❛t❛♠❡♥t❡ ✶✵✵ ❡str❛❞❛s ♥❡ss❡ r❡✐♥❛❞♦✱ ✉♠❛ ✈❡③ q✉❡ ♦ ♥ú♠❡r♦ ❞❡ ❡str❛❞❛s ♣r❡❝✐s❛ s❡r
❈❛♣ít✉❧♦ ✶✳ ❚❊❖❘■❆ ❉❖❙ ●❘❆❋❖❙ ✶✳✶✳ ●r❛✉ ❞♦ ✈ért✐❝❡
✉♠ ♠ú❧t✐♣❧♦ ❞❡ ✸✳ P♦❞❡r✐❛♠ ❡①✐st✐r ✾✾ ❡str❛❞❛s ❡✱ ❛ss✐♠✱ t❡rí❛♠♦s ✻✻ ❝✐❞❛❞❡s ♥❡ss❡ r❡✐♥❛❞♦✳
❊①❡♠♣❧♦ ✺ ❊♠ ■♥t❡r✐♦r❛♥❛ ❡①✐st❡♠ ✶✺ t❡❧❡❢♦♥❡s✳ ❊❧❡s ♣♦❞❡♠ s❡r ❝♦♥❡❝t❛❞♦s ❞❡ ♠♦❞♦ q✉❡✿
✭❛✮ ❝❛❞❛ t❡❧❡❢♦♥❡ s❡❥❛ ❝♦♥❡❝t❛❞♦ ❛ ❡①❛t❛♠❡♥t❡ ✼ ♦✉tr♦s❀
✭❜✮✹ t❡❧❡❢♦♥❡s ❡stã♦ ❝♦♥❡❝t❛❞♦ ❛ ✸ ♦✉tr♦s✱ ✽ t❡❧❡❢♦♥❡s ❡stã♦ ❝♦♥❡❝t❛❞♦s ❛ ✻ ♦✉tr♦s ❡ ✸ t❡❧❡❢♦♥❡s ❡stã♦ ❝♦♥❡❝t❛❞♦s ❛ ✺ ♦✉tr♦s❄
✭❛✮ ❚❡♠♦s ✶✺ t❡❧❡❢♦♥❡s✱ ❝❛❞❛ ✉♠ ❧✐❣❛❞♦s ❛ ✼ ♦✉tr♦s✳ ❙❡ ✐ss♦ ❢♦ss❡ ♣♦ssí✈❡❧✱ t❛♠❜é♠ s❡r✐❛ ♣♦ssí✈❡❧ ✉♠ ❣r❛❢♦ ❝♦♠ ✶✺ ✈ért✐❝❡s✱ ❝❛❞❛ ✉♠ ❝♦♠ ❣r❛✉ ✼✳ P♦rt❛♥t♦✱ ♦ ♥ú♠❡r♦s ❞❡ ❛r❡st❛s é (15×7)/2✳ ❚♦❞❛✈✐❛✱ ❡ss❡ ♥ú♠❡r♦ ♥ã♦ é ✐♥t❡✐r♦✳ ❉❡ss❛ ❢❡✐t❛✱ ❛ ❝♦♥✜❣✉r❛çã♦
t❡❧❡❢ô♥✐❝❛ ❞✐s❝r✐♠✐♥❛❞❛ ♥♦ ✐t❡♠ ❛ é ✐♠♣♦ssí✈❡❧ ❞❡ s❡r r❡❛❧✐③❛❞❛✳
✭❜✮ ❙✉♣♦♥❤❛♠♦s q✉❡ s❡❥❛ ♣♦ssí✈❡❧ ✉♠ ❣r❛❢♦✱ ♥♦ q✉❛❧✿ ✹ ✈ért✐❝❡s t❡♥❤❛♠ ❣r❛✉ ✸❀ ✽ t❡♥❤❛♠ ❣r❛✉ ✻❀ ❡ ✸ t❡♥❤❛♠ ❣r❛✉ ✺✳ P♦❞❡♠♦s s♦♠❛r ♦s ❣r❛✉s ❞❡st❡ ❣r❛❢♦✱ ❝♦♥❢♦r♠❡ ❛❞✐❛♥t❡✿
4.3 + 8.6 + 3.5 = 12 + 48 + 15 = 75,
♥♦ ❡♥t❛♥t♦✱ ♣❡❧♦ t❡♦r❡♠❛ ✶✱ t❡♠♦s 2×E = 75. ❯♠❛ ❝♦♥tr❛❞✐çã♦✱ ♣♦✐s ❊❂✼✺✴✷ ♥ã♦ é
✉♠ ♥ú♠❡r♦ ✐♥t❡✐r♦✳ ◆♦t❡ q✉❡✱ ❡ss❡ ❣r❛❢♦ ♣♦ss✉✐r✐❛ ✼ ✈ért✐❝❡s ❞❡ ❣r❛✉ í♠♣❛r✱ ♦✉ s❡❥❛✱ ✉♠ ♥ú♠❡r♦ í♠♣❛r ❞❡ ✈ért✐❝❡s ❞❡ ❣r❛✉ í♠♣❛r✱ ❝♦♥tr❛r✐❛♥❞♦ ♦ ❝♦r♦❧ár✐♦ ❛❜❛✐①♦✳
❈♦r♦❧ár✐♦ ✶ ❖ ♥ú♠❡r♦ ❞❡ ✈ért✐❝❡s í♠♣❛r❡s ❡♠ ✉♠ ❣r❛❢♦ q✉❛❧q✉❡r t❡♠ q✉❡ s❡r ♣❛r✳ ❉❡♠♦♥str❛çã♦✿ ❙✉♣♦♥❤❛ q✉❡ ✐ss♦ ♥ã♦ ♦❝♦rr❡✱ ❞✐❣♦✱ ♦ ♥ú♠❡r♦ ❞❡ ✈ért✐❝❡s í♠♣❛r❡s ♣♦❞❡ s❡r í♠♣❛r✳ ❙❡♥❞♦ ❛ss✐♠✱ ❛ s♦♠❛ ❞♦s ❣r❛✉s ❞♦s ✈ért✐❝❡s s❡r✐❛ í♠♣❛r✳ ❚♦❞❛✈✐❛✱ ❡ss❛ s♦♠❛ é ✐❣✉❛❧ ❛ ❞✉❛s ✈❡③❡s ♦ ♥ú♠❡r♦s ❞❡ ❛r❡st❛s✭2×E✮✱ q✉❡ é ♣❛r❀ ✉♠❛ ❝♦♥tr❛❞✐çã♦✳
❆♣ós ❡ss❡ ❝♦r♦❧ár✐♦✱ ❜❛st❛ ♦❜s❡r✈❛r ♥❛ q✉❡stã♦ ❛♥t❡r✐♦r q✉❡ ♦ ♥ú♠❡r♦ ❞❡ ✈ért✐❝❡s í♠♣❛r❡s s❡r✐❛ í♠♣❛r ❡♠ ❝❛❞❛ ✉♠ ❞♦s ❣r❛❢♦s❀ ❞❡st❛ ❢❡✐t❛✱ ♥ã♦ é ♣♦ssí✈❡❧ ❡ss❡s ❣r❛❢♦s ❡①✐st✐r❡♠✳
❈❛♣ít✉❧♦ ✶✳ ❚❊❖❘■❆ ❉❖❙ ●❘❆❋❖❙ ✶✳✶✳ ●r❛✉ ❞♦ ✈ért✐❝❡
❊①❡♠♣❧♦ ✻ ❯♠ r❡✐ t❡♠ ✶✾ ✈❛ss❛❧♦s✳ ➱ ♣♦ssí✈❡❧ q✉❡ ❝❛❞❛ ✈❛ss❛❧♦ t❡♥❤❛ ♦✉ ✶✱ ♦✉ ✺✱ ♦✉ ✾ ✈❛ss❛❧♦s ✈✐③✐♥❤♦s❄
P♦❞❡rí❛♠♦s ♣❡r❣✉♥t❛r s❡ é ♣♦ssí✈❡❧ ✉♠ ❣r❛❢♦ ❝♦♠ ✶✾ ✈ért✐❝❡s✱ ❞❡ ❢♦r♠❛ q✉❡ ❝❛❞❛ ✈ért✐❝❡ t❡♥❤❛ ❣r❛✉ ✐❣✉❛❧ ♦✉ ❛ ✶✱♦✉ ❛ ✺✱ ♦✉ ❛ ✾❄ ❊♠ q✉❛❧q✉❡r ❞♦s ✸ ❝❛s♦s✱ t❡♠♦s ✈ért✐❝❡s í♠♣❛r❡s✳ ❆ss✐♠✱ ❡ss❡ ❣r❛❢♦ ♣♦ss✉✐r✐❛ ✉♠ ♥ú♠❡r♦ í♠♣❛r ❞❡ ✈ért✐❝❡s í♠♣❛r❡s✳ ▲♦❣♦✱ ♥ã♦ é ♣♦ssí✈❡❧✳
❊①❡♠♣❧♦ ✼ ❏♦ã♦ ✈♦❧t❛♥❞♦ ❞❛ ❉✐s♥❡✐❧â♥❞✐❛✱ ❞✐ss❡ q✉❡ ✈✐✉ ✉♠ ❧❛❣♦ ❡♥❝❛♥t❛❞♦ ❝♦♠ ✼ ✐❧❤❛s✱ ❝❛❞❛ ✉♠❛ ❞❡❧❛s t❡♥❞♦ ✶✱ ✸ ♦✉ ✺ ♣♦♥t❡s ❝❤❡❣❛♥❞♦ ❛ ❡❧❛s✳ ➱ ✈❡r❞❛❞❡ q✉❡ ♣❡❧♦ ♠❡♥♦s ✉♠❛ ❞❡ss❛s ♣♦♥t❡s t❡♠ q✉❡ ❧❡✈❛r ♣❛r❛ t❡rr❛ ✜r♠❡❄
❙✐♠✳ ❆s s❡t❡ ✐❧❤❛s ❡ ❛ t❡rr❛ ✜r♠❡ sã♦ ♦s ✈ért✐❝❡s ❡ ❛s ♣♦♥t❡s✱ ❛s ❛r❡st❛s✳ ❙❡ ✉♠❛ ❞❛s ♣♦♥t❡s ♥ã♦ ❢♦ss❡ ♣❛r❛ t❡rr❛ ✜r♠❡✱ t❡rí❛♠♦s ✉♠ ❣r❛❢♦ ❝♦♠ ♥ú♠❡r♦ í♠♣❛r ❞❡ ✈ért✐❝❡s ❞❡ ❣r❛✉ í♠♣❛r✳ ❆♦ ❝♦♥s✐❞❡r❛r♠♦s ❛ t❡rr❛ ✜r♠❡ ❝♦♠♦ ✉♠ ✈ért✐❝❡✱ t❡r❡♠♦s ✉♠ ♥ú♠❡r♦ ♣❛r ❞❡ ✈ért✐❝❡s í♠♣❛r❡s✳ ◆❛ ✜❣✉r❛ ✶✳✺✱ r❡♣r❡s❡♥t❛r❡♠♦s ♣♦r ❣r❛❢♦ ✉♠❛ ♣♦ssí✈❡❧ ❝♦♥❞✐çã♦ ❞❛s ✐❧❤❛s✿
❋✐❣✉r❛ ✶✳✺✿ ●r❛❢♦ ❞♦ ❡①❡♠♣❧♦ ✼
❆s ✐❧❤❛s sã♦ ♦s ✈ért✐❝❡s ♥✉♠❡r❛❞♦s ❞❡ ✶ ❛ ✼ ❡ ❛ t❡rr❛ ✜r♠❡ é ♦ ✽✳ ◆♦t❡ q✉❡ ❞❡ ❝❛❞❛ ✐❧❤❛ s❛❡♠ ✶✱ ✸ ♦✉ ✺ ♣♦♥t❡s✱ ❝♦♠♦ ❞❡t❡r♠✐♥❛ ♦ ❡♥✉♥❝✐❛❞♦✳ ▼❛s✱ ❡①✐st❡♠ ♠✉✐t❛s ♦✉tr❛s ❢♦r♠❛s ❞❡ r❡♣r❡s❡♥t❛r ♦ ♣r♦❜❧❡♠❛ tr❛t❛❞♦ ♣♦r ❣r❛❢♦✳ ❊ss❛ ✐❧✉str❛çã♦ é ❛♣❡♥❛s ✉♠ ♠♦❞❡❧♦ ❡s❝♦❧❤✐❞♦✳ ◆♦ ❡♥t❛♥t♦✱ ❡♠ q✉❛❧q✉❡r r❡♣r❡s❡♥t❛çã♦ ✉t✐❧✐③❛❞❛✱ ♣❡❧♦ ♠❡♥♦s ✉♠❛ ♣♦♥t❡ t❡rá q✉❡ ❧❡✈❛r ❛ t❡rr❛ ✜r♠❡✳
❈❛♣ít✉❧♦ ✶✳ ❚❊❖❘■❆ ❉❖❙ ●❘❆❋❖❙ ✶✳✷✳ ●r❛❢♦s ❝♦♥❡①♦s
❊①❡♠♣❧♦ ✽ Pr♦✈❡ q✉❡ ♦ ♥ú♠❡r♦ ❞❡ ♣❡ss♦❛s q✉❡ ✈✐✈❡r❛♠ ♥❛ ❚❡rr❛ ❡ q✉❡ ❛♣❡rt❛r❛♠ ♠ã♦s ✉♠ ♥ú♠❡r♦ í♠♣❛r ❞❡ ✈❡③❡s é ♣❛r✳
❙✉♣♦♥❤❛ q✉❡ ✐ss♦ ♥ã♦ s❡❥❛ ✈❡r❞❛❞❡✱ ♣♦r ❝♦♥s❡❣✉✐♥t❡✱ ♦ ♥ú♠❡r♦ ❞❡ ♣❡ss♦❛s q✉❡ ✈✐✈❡r❛♠ ♥❛ ❚❡rr❛ ❡ ❛♣❡rt❛r❛♠ ♠ã♦s ✉♠ ♥ú♠❡r♦ í♠♣❛r ❞❡ ✈❡③❡s é í♠♣❛r✳ ▲♦❣♦✱ s❡r✐❛ ♣♦ssí✈❡❧ ✉♠ ❣r❛❢♦ ❝✉❥♦ ♦s ✈ért✐❝❡s r❡♣r❡s❡♥t❛r✐❛♠ ❛s ♣❡ss♦❛s ❡ ❛s ❛r❡st❛s✱ ♦s ❛♣❡rt♦s ❞❡ ♠ã♦s✳ ❆ss✐♠✱ ♦ ♥ú♠❡r♦ ❞❡ ❛♣❡rt♦s ❞❡ ♠ã♦s s❡r✐❛ ♦ ❣r❛✉ ❞❡ ❝❛❞❛ ✈ért✐❝❡ ❡ t❡rí❛♠♦s ✉♠ ❣r❛❢♦ ❝♦♠ ✉♠ ♥ú♠❡r♦ í♠♣❛r ❞❡ ✈ért✐❝❡s ❞❡ ❣r❛✉ í♠♣❛r✱ ♦ q✉❡ ❝♦♥tr❛❞✐③ ♦ ❝♦r♦❧ár✐♦ ✶✳
✶✳✷ ●r❛❢♦s ❝♦♥❡①♦s
❱❡r❡♠♦s ❛❧❣✉♠❛s ❞❡✜♥✐çõ❡s ♥❡❝❡ssár✐❛s✿
❉❡✜♥✐çã♦ ✺ ❯♠ ♣❛ss❡✐♦ é ✉♠❛ s❡q✉ê♥❝✐❛ ❞❡ ❛r❡st❛s ❞♦ t✐♣♦v0v1, v1v2, v2v3, ..., v(s−1)vs.
◆♦ ❝❛s♦ ❡♠ q✉❡ t♦❞❛s ❛s ❛r❡st❛s sã♦ ❞✐st✐♥t❛s✱ ♦ ♣❛ss❡✐♦ é ✉♠❛ ❚r✐❧❤❛✳ ❘❡❝❡❜❡♥❞♦ ♦ ♥♦♠❡ ❞❡ ❚r✐❧❤❛ ❢❡❝❤❛❞❛✱ ❝❛s♦ v0 =vs✳
❉❡✜♥✐çã♦ ✻ ❈❛♠✐♥❤♦ é ✉♠ ♣❛ss❡✐♦ ♦♥❞❡ t♦❞❛s ❛s ❛r❡st❛s ❡ t♦❞♦s ♦s ✈ért✐❝❡s sã♦ ❞✐st✐♥t♦s✱ ♣♦❞❡♥❞♦ ❝♦✐♥❝✐❞✐r ❛♣❡♥❛s ♦ ♣r✐♠❡✐r♦ ❡ ♦ ú❧t✐♠♦✳ ❙❡ ✐ss♦ ♦❝♦rr❡r✱ t❡r❡♠♦s ✉♠ ❈✐❝❧♦ ✖ ✉♠ ❝❛♠✐♥❤♦ ❢❡❝❤❛❞♦✳
➱ s❛❧✉t❛r r❡ss❛❧t❛r q✉❡ ❡①✐st❡♠ ❞❡✜♥✐çõ❡s ❞✐✈❡r❣❡♥t❡s ❞❡ ♣❛ss❡✐♦ ❡ ❝❛♠✐♥❤♦ ♥❛ ❧✐t❡r❛t✉r❛✱ ❝♦♥❢♦r♠❡ ❝✐t❛♠♦s ♥❛ ✐♥tr♦❞✉çã♦✳
◆❛ ✜❣✉r❛ ✶✳✻✱ t❡♠♦s ❞✐✈❡rs♦s ❝✐❝❧♦s✱ ✉♠ ❞❡❧❡s✿ ❆❇❊❆✱ ✐♥❞✐❝❛❞♦ ♣❡❧♦s ✈ért✐❝❡s❀ ❡ tr✐❧❤❛s ❢❡❝❤❛❞❛s✱ ♣♦r ❡①❡♠♣❧♦✿ ❆❇✱ ❇❊✱ ❊❉✱ ❉❈✱ ❈❊✱ ❊❆✳
❉❡✜♥✐çã♦ ✼ ❈✐♥t✉r❛ ❞❡ ✉♠ ❣r❛❢♦ ● é ♦ ♥ú♠❡r♦ ❞❡ ✈ért✐❝❡s ❞♦ ♠❡♥♦r ❝✐❝❧♦ ❞❡ ●✳ ❘❡♣r❡s❡♥t❛r❡♠♦s ❛ ❝✐♥t✉r❛ ♣♦r ❣✳
◆♦ ❝❛s♦ ❞♦ ❣r❛❢♦ ❞❛ ✜❣✉r❛ ✶✳✻✱ t❡♠♦s g = 3✳
❉❡✜♥✐çã♦ ✽ ❯♠ ❣r❛❢♦ é ❞✐t♦ ❝♦♥❡①♦ s❡✱ ❞❛❞♦s ❞♦✐s ✈ért✐❝❡s q✉❛✐sq✉❡r✱ ❡①✐st❡ ✉♠ ❝❛♠✐♥❤♦ ❧✐❣❛♥❞♦ ♦s ✈ért✐❝❡s✳ ❯♠ ❣r❛❢♦ q✉❡ ♥ã♦ é ❝♦♥❡①♦✱ é ❞✐t♦ ❞❡s❝♦♥❡①♦✳
❈❛♣ít✉❧♦ ✶✳ ❚❊❖❘■❆ ❉❖❙ ●❘❆❋❖❙ ✶✳✷✳ ●r❛❢♦s ❝♦♥❡①♦s
❋✐❣✉r❛ ✶✳✻✿ ❈✐❝❧♦s ❡ tr✐❧❤❛s
◆♦ ♣r♦❜❧❡♠❛ ✷✱ ❝♦♠❡♥t❛♠♦s q✉❡ ♦ ❣r❛❢♦ ❞❛ ✜❣✉r❛ ✶✳✸ ❡r❛ ❞❡s❝♦♥❡①♦✳
❈♦♠♦ ♣♦❞❡♠♦s r❡✈❡r✱ ♥ã♦ é ♣♦ssí✈❡❧ ✐r ❞♦ ✈ért✐❝❡ ✾ ❛♦ ✈ért✐❝❡ ✹✱ ✉♠❛ ✈❡③ q✉❡ ♥ã♦ ❡①✐st❡ ✉♠ ❝❛♠✐♥❤♦ ❧✐❣❛♥❞♦ ❡ss❡s ✈ért✐❝❡s✳ P♦❞❡rí❛♠♦s ♦❧❤❛r ♣❛r❛ ❡ss❡ ❣r❛❢♦ ❡ ✐♥t❡r♣r❡tá✲ ❧♦ ❝♦♠♦ s❡♥❞♦ ✷ ❣r❛❢♦s✱ ♠❛s✱ ♥♦ ❝❛s♦ ❞❛ s♦❧✉çã♦ ❞♦ ♥♦ss♦ ♣r♦❜❧❡♠❛✱ t❡♠♦s ✉♠ ú♥✐❝♦ ❣r❛❢♦✱ ❝✉❥❛s ❧✐❣❛çõ❡s r❡♣r❡s❡♥t❛♠ q✉❡ ❡①✐st❡ ✈♦♦ ❞✐r❡t♦ ❡♥tr❡ ❛s ❝✐❞❛❞❡s✱ ❡ ❛ ❛✉sê♥❝✐❛ ❞❡ ❛r❡st❛s ❧✐❣❛♥❞♦ ♦s ✈ért✐❝❡s s✐❣♥✐✜❝❛ q✉❡ ♥ã♦ ❡①✐st❡ ✈♦♦ ❞✐r❡t♦ ❡♥tr❡ ❛s ❝✐❞❛❞❡s✳ ❏á ❞✐ss❡♠♦s q✉❡ ❡ss❡ ❣r❛❢♦ ♥ã♦ é ❝♦♥❡①♦❀ ♣♦ré♠✱ ❡❧❡ é ❢♦r♠❛❞♦ ♣♦r ❞♦✐s ✧❢r❛❣♠❡♥t♦s✧✱ ❡ ❡♠ ❝❛❞❛ ✉♠ ❞❡❧❡s é ♣♦ssí✈❡❧ ✐r ❞❡ ✉♠ ✈ért✐❝❡ ❛ q✉❛❧q✉❡r ♦✉tr♦✳ ❊ss❡s ❢r❛❣♠❡♥t♦s sã♦ ❛s ❝♦♠♣♦♥❡♥t❡s ❝♦♥❡①❛s ❞❡ ✉♠ ❣r❛❢♦✳ ❯♠ ❣r❛❢♦ ❢♦r♠❛❞♦ ♣♦r ❛♣❡♥❛s ✉♠❛ ❝♦♠♣♦♥❡♥t❡ ❝♦♥❡①❛ é ✉♠ ❣r❛❢♦ ❝♦♥❡①♦✳
❖ ❝♦♥❝❡✐t♦ ❞❡ ❝♦♥❡①✐❞❛❞❡ é ❞❡ ♠✉✐t❛ ✉t✐❧✐❞❛❞❡ ♥♦s ❣r❛❢♦s✳ ▼❛s✱ ❝♦♠♦ ♣♦❞❡♠♦s s❛❜❡r s❡ ✉♠ ❣r❛❢♦ é ♦✉ ♥ã♦ ❝♦♥❡①♦❄ ❆♣r❡s❡♥t❛r❡♠♦s ✉♠❛ ♣r♦♣♦s✐çã♦ q✉❡ ❣❛r❛♥t❡ ❛ ❝♦♥❡①✐❞❛❞❡ ❞❡ ✉♠ ❣r❛❢♦✳
Pr♦♣♦s✐çã♦ ✶ ❙❡ ● é ✉♠ ❣r❛❢♦ ❝♦♠ ♥ ✈ért✐❝❡s✱ t♦❞♦s ❝♦♠ ❣r❛✉ ♠❛✐♦r ♦✉ ✐❣✉❛❧ ❛ (n−1) 2 ✱ ● é ❝♦♥❡①♦✳
❈❛♣ít✉❧♦ ✶✳ ❚❊❖❘■❆ ❉❖❙ ●❘❆❋❖❙ ✶✳✷✳ ●r❛❢♦s ❝♦♥❡①♦s
❉❡♠♦♥str❛çã♦✿ ❙✉♣♦♥❤❛ q✉❡ ● ♥ã♦ é ❝♦♥❡①♦✳ ▲♦❣♦ ❡①✐st❡ ♣❡❧♦ ♠❡♥♦s ✷ ✈ért✐❝❡s ❞❡ ● q✉❡ ♥ã♦ ❡stã♦ ❧✐❣❛❞♦s ♣♦r ♥❡♥❤✉♠ ❝❛♠✐♥❤♦✳ P♦r ❤✐♣ót❡s❡✱ ❞❡ ❝❛❞❛ ✉♠ ❞❡ss❡s ✈ért✐❝❡s ✐♥❝✐❞❡ ♣❡❧♦ ♠❡♥♦s (n−1)
2 ❛r❡st❛s✳ ◆♦ ❡♥t❛♥t♦✱ ♥❡♥❤✉♠❛ ❞❡❧❛s ♣♦❞❡ ❡stá ❧✐❣❛❞❛ ❛♦ ♠❡s♠♦ ✈ért✐❝❡✱ ❝❛s♦ ✐ss♦ ♦❝♦rr❡ss❡✱ ❡①✐st✐r✐❛ ✉♠ ❝❛♠✐♥❤♦ ❧✐❣❛♥❞♦ ♦s ✷ ✈ért✐❝❡s✳ ❆ss✐♠✱ ❝❛❞❛ ✉♠❛ ❞❡ss❛s ❛r❡st❛s ❡stá ❧✐❣❛❞❛ ❛ ✉♠ ✈ért✐❝❡ ❞✐st✐♥t♦✳ ❈♦♥s❡q✉❡♥t❡♠❡♥t❡✱ ♦ ♥ú♠❡r♦ ❞❡ ✈ért✐❝❡s ❞♦ ❣r❛❢♦ é
(n−1)
2 +
(n−1)
2 + 2 =n+ 1.
❯♠❛ ❝♦♥tr❛❞✐çã♦✱ ♣♦✐s ♦ ❣r❛❢♦ t❡♠ ♥ ✈ért✐❝❡s✳ ❆ ❝♦♥tr❛❞✐çã♦ é ❞❡❝♦rr❡♥t❡ ❞❛ ❤✐♣ót❡s❡ ❞♦ ❣r❛❢♦ s❡r ❞❡s❝♦♥❡①♦✳ P♦r ❝♦♥s❡❣✉✐♥t❡✱ ● é ❝♦♥❡①♦✳
✶✳✷✳✶ ●r❛❢♦ ❊✉❧❡r✐❛♥♦
■♥✐❝✐❛r❡♠♦s ❡ss❛ s✉❜s❡çã♦ ❝♦♠ ✉♠ ♣r♦❜❧❡♠❛ q✉❡ ✜❝♦✉ ♠✉✐t♦ ❢❛♠♦s♦✱ s❡♥❞♦ ❝♦♥s✐❞❡r❛❞♦ ♦ ♣r✐♠❡✐r♦ r❡❧❛t♦ ❞❛ t❡♦r✐❛ ❞♦s ❣r❛❢♦s✳ ❊♠❜♦r❛ ❡st❛ t❡♦r✐❛ t❡♥❤❛ s❡ ❞❡s❡♥✈♦❧✈✐❞♦ só ♥♦ sé❝✉❧♦ ♣❛ss❛❞♦✱ ❡st❡ r❡❧❛t♦ é ❞♦ sé❝✉❧♦ ❳■■■✳ ❊ ♥♦s ❛❥✉❞❛rá ❛ ❡♥t❡♥❞❡r ❛ ❞❡✜♥✐çã♦ ❞❡ ❣r❛❢♦ ❊✉❧❡r✐❛♥♦✳
❊①❡♠♣❧♦ ✾ ❆ ✜❣✉r❛ ✶✳✼ ♠♦str❛ ❛ ❝✐❞❛❞❡ ❞❡ ❑ö♥✐❣s❜❡r❣✳ ❆ ❝✐❞❛❞❡ é ❝♦rt❛❞❛ ♣♦r ✉♠ r✐♦ q✉❡ t❡♠ ❞✉❛s ✐❧❤❛s✳ ❊①✐st❡♠ s❡t❡ ♣♦♥t❡s ❧✐❣❛♥❞♦ ❛s ❞✐✈❡rs❛s ♣❛rt❡s ❞❛ ❝✐❞❛❞❡✳ ➱ ♣♦ssí✈❡❧ ♣❛ss❡❛r ♣❡❧❛ ❝✐❞❛❞❡ ❝r✉③❛♥❞♦ ❝❛❞❛ ♣♦♥t❡ ❡①❛t❛♠❡♥t❡ ✉♠❛ ✈❡③❄ ✸
❊ss❡ ♣r♦❜❧❡♠❛ ❢♦✐ r❡s♦❧✈✐❞♦ ♣♦r ❊✉❧❡r ❡♠ ✶✼✸✻✳ ❆s ✐❧❤❛s ❡ ♦s ❞♦✐s ❧❛❞♦s ❞❛ ❝✐❞❛❞❡ ♣♦❞❡♠ s❡r ✈✐st♦s ❝♦♠♦ ♦s ✈ért✐❝❡s ❞♦ ❣r❛❢♦ ❡ ❛s ♣♦♥t❡s ❝♦♠♦ ❛s ❛r❡st❛s✳ ◆♦t❡ q✉❡ ♣❛r❛ ♣❛ss❡❛r♠♦s ♣♦r ❝❛❞❛ ♣♦♥t❡ ✉♠❛ ú♥✐❝❛ ✈❡③✱ ♣r❡❝✐s❛rí❛♠♦s ❞❡ ✉♠ ♥ú♠❡r♦ ♣❛r ❞❡ ❛r❡st❛s✭♣♦♥t❡s✮ ✐♥❝✐❞❡♥t❡s ❡♠ ❝❛❞❛ ✈ért✐❝❡✱ ♣♦✐s t❡rí❛♠♦s q✉❡ ❡♥tr❛r ❡ s❛✐r ❞❡ ❝❛❞❛ ✈ért✐❝❡ ✭✐❧❤❛ ♦✉ ♣❛rt❡ ❞❛ ❝✐❞❛❞❡✮ ♣♦r ❛r❡st❛s ❞✐st✐♥t❛s✳ ◆♦ ❡♥t❛♥t♦✱ ❝♦♠♦ ♣♦❞❡♠♦s ✈❡r ♥♦ ❣r❛❢♦ ❞❛ ✜❣✉r❛ ✶✳✽✱ q✉❡ ♠♦❞❡❧❛ ♦ ♣r♦❜❧❡♠❛✱ ❡♠ ❝❛❞❛ ✈ért✐❝❡ ✐♥❝✐❞❡ ✉♠ ♥ú♠❡r♦ í♠♣❛r ❞❡ ❛r❡st❛s✱ ♦ q✉❡ t♦r♥❛ ♦ ♣r♦❜❧❡♠❛ ✐♠♣♦ssí✈❡❧✳
✸ ❖ ♣r✐♠❡✐r♦ ❡ ♠❛✐s ❢❛♠♦s♦ ♣r♦❜❧❡♠❛ ❡♠ ❚❡♦r✐❛ ❞♦s ●r❛❢♦s✱ r❡s♦❧✈✐❞♦ ♣♦r ❊✉❧❡r ❡♠ ✶✼✸✻✳ ❊✉❧❡r
tr❛♥s❢♦r♠♦✉ ♦ ♣r♦❜❧❡♠❛ ❡♠ ✉♠ ♣r♦❜❧❡♠❛ ❞❡ ❣r❛❢♦s✳
❈❛♣ít✉❧♦ ✶✳ ❚❊❖❘■❆ ❉❖❙ ●❘❆❋❖❙ ✶✳✷✳ ●r❛❢♦s ❝♦♥❡①♦s
❋✐❣✉r❛ ✶✳✼✿ P♦♥t❡s ❞❡ ❑ö♥✐❣s❜❡r❣
❋♦♥t❡✿ ❖▲■❱❊■❘❆✱ ✷✵✶✷
❋✐❣✉r❛ ✶✳✽✿ ●r❛❢♦ ❞❛s ♣♦♥t❡s ❞❞❡ ❑ö✐♥✐❣s❜❡r❣
❋♦♥t❡✿ ❊❧❛❜♦r❛❞♦ ♣❡❧❛ ❛✉t♦r❛
❱❡❥❛♠♦s s❡ ❝♦♥s❡❣✉✐♠♦s ❞❡s❡♥❤❛r ♦s ●r❛❢♦s ❛ s❡❣✉✐r✱ s❡♠ r❡t✐r❛r ♦ ❧á♣✐s ❞♦ ♣❛♣❡❧✱ ❞❡s❡♥❤❛♥❞♦ ❝❛❞❛ ❛r❡st❛ ✉♠❛ ú♥✐❝❛ ✈❡③✳
P❛r❛ ❞❡s❡♥❤❛r ♦ ❣r❛❢♦ ✶✱ ❜❛st❛ s❡❣✉✐r ❛ s❡q✉ê♥❝✐❛ ❆❇❉❈❆❉ ✭é ✉♠❛ ❞❛s ❢♦r♠❛s✮ ✖ ♥♦t❡ q✉❡ ✐♥✐❝✐❛♠♦s ❡ t❡r♠✐♥❛♠♦s ❡♠ ✈ért✐❝❡s ❞✐st✐♥t♦s❀ ❥á ♦ ❣r❛❢♦ ✷✱ ♣♦r ♠❛✐s q✉❡ t❡♥t❡♠♦s ❞❡ ❞✐❢❡r❡♥t❡s ❢♦r♠❛s ♥ã♦ ❝♦♥s❡❣✉✐♠♦s❀ ♦ ●r❛❢♦ ✸✱ ❡♥tr❡ ❛s ✐♥ú♠❡r❛s ♣♦ss✐❜✐❧✐❞❛❞❡s✱ ♣♦❞❡♠♦s s❡❣✉✐r ❛ s❡q✉ê♥❝✐❛ ❞♦s ✈ért✐❝❡s✿ ◆■❏❑▲❏▼▲■▼◆ ✭♥❡st❡ ❣r❛❢♦✱ ✐♥✐❝✐❛♠♦s ❡ t❡r♠✐♥❛♠♦s ♥♦ ♠❡s♠♦ ✈ért✐❝❡✮✳ ❘❡s♦❧✈❡♥❞♦ ❡ss❡ ♣r♦❜❧❡♠❛✱ ♣♦❞❡♠♦s ♣❡r❝❡❜❡r q✉❡✿
❈❛♣ít✉❧♦ ✶✳ ❚❊❖❘■❆ ❉❖❙ ●❘❆❋❖❙ ✶✳✷✳ ●r❛❢♦s ❝♦♥❡①♦s
❋✐❣✉r❛ ✶✳✾✿ ●r❛❢♦s ✶✱ ✷ ❡ ✸
P❛r❛ q✉❡ ✉♠ ❣r❛❢♦ s❡❥❛ ♣❡r❝♦rr✐❞♦ s❡♠ ❧❡✈❛♥t❛r ♦ ❧á♣✐s ❞♦ ♣❛♣❡❧✱ ❞❡ ❢♦r♠❛ q✉❡ ❝❛❞❛ ❛r❡st❛ só s❡❥❛ ❞❡s❡♥❤❛❞❛ ✉♠❛ ú♥✐❝❛ ✈❡③✱ ❡❧❡ ♥ã♦ ♣♦❞❡ t❡r ♠❛✐s q✉❡ ✷ ✈ért✐❝❡s í♠♣❛r❡s✳ ❖ ❣r❛❢♦ ✶ ♣♦ss✉✐ ❛♣❡♥❛s ✷ ✈ért✐❝❡s í♠♣❛r❡s✱ ♣♦r ✐ss♦ ❝♦♥s❡❣✉✐♠♦s ❞❡s❡♥❤á✲❧♦✱ ❥á ♦ ❣r❛❢♦ ✷ ♣♦ss✉✐ ✹ ✈ért✐❝❡s í♠♣❛r❡s✱ ♥ã♦ s❡♥❞♦ ♣♦ssí✈❡❧ ❞❡s❡♥❤á✲❧♦ s♦❜ ❡ss❛s ❝♦♥❞✐çõ❡s❀
P❛r❛ q✉❡ s❡❥❛ ♣♦ssí✈❡❧ ♣❡r❝♦rr❡r ♦ ❣r❛❢♦ ♥❡ss❛s ♠❡s♠❛s ❝♦♥❞✐çõ❡s ❡ ✈♦❧t❛r ❛♦ ✈ért✐❝❡ ✐♥✐❝✐❛❧✱ é ♥❡❝❡ssár✐♦ q✉❡ t♦❞♦s ♦s ✈ért✐❝❡s t❡♥❤❛♠ ❣r❛✉ ♣❛r ✖ ❝r✐tér✐♦ ✉t✐❧✐③❛❞♦ ♣♦r ❊✉❧❡r ♥♦ ♣r♦❜❧❡♠❛ ❞❛s P♦♥t❡s ❞❡ ❑ö♥✐❣s❜❡r❣✳ P❡r❝❡❜❛♠♦s q✉❡ ♥♦ ❣r❛❢♦ ✸ t♦❞♦s ♦s ✈ért✐❝❡s t❡♠ ❣r❛✉ ♣❛r✳
❉❡✜♥✐çã♦ ✾ ❯♠ ❣r❛❢♦ ❝♦♥❡①♦ ❝♦♠ ♥ ❛r❡st❛s é ❡✉❧❡r✐❛♥♦ q✉❛♥❞♦ ♣♦ss✉✐ ✉♠❛ tr✐❧❤❛ ❢❡❝❤❛❞❛ ✖ ✉♠ ❝✐r❝✉✐t♦ ✖ ❞❡ ❝♦♠♣r✐♠❡♥t♦ ♥✱ ♦✉ s❡❥❛✱ q✉❛♥❞♦ ♣❡r❝♦rr❡♠♦s t♦❞❛s ❛s ❛r❡st❛s ✉♠❛ ú♥✐❝❛ ✈❡③✱ t❡r♠✐♥❛♥❞♦ ♥♦ ✈ért✐❝❡ ✐♥✐❝✐❛❧✳ ◗✉❛♥❞♦ ♦ ❣r❛❢♦ ♣♦ss✉✐ ✉♠❛ tr✐❧❤❛ ❛❜❡rt❛ ❞❡ ❝♦♠♣r✐♠❡♥t♦ ♥ ✲ ❞✐❣♦✱ q✉❛♥❞♦ ♣❡r❝♦rr❡♠♦s t♦❞❛s ❛s ❛r❡st❛s ❞♦ ❣r❛❢♦ ✉♠❛ ú♥✐❝❛ ✈❡③✱ ♠❛s ♥ã♦ t❡r♠✐♥❛♠♦s ♥♦ ✈ért✐❝❡ ✐♥✐❝✐❛❧ ✲ ❞✐③❡♠♦s q✉❡ ♦ ❣r❛❢♦ é s❡♠✐❡✉❧❡r✐❛♥♦✳
❖ ❣r❛❢♦ ✸✱ ❞❛ ✜❣✉r❛ ✶✳✾ é ❡✉❧❡r✐❛♥♦✱ ♣♦✐s ♣♦ss✉✐ ✉♠❛ tr✐❧❤❛ ❢❡❝❤❛❞❛❀ ❡ ♦ ❣r❛❢♦ ✶ ❞❛ ♠❡s♠❛ ✜❣✉r❛ é s❡♠✐❡✉❧❡r✐❛♥♦✳
❊①❡♠♣❧♦ ✶✵ ❯♠ ❣r✉♣♦ ❞❡ ✐❧❤❛s ❡stã♦ ❧✐❣❛❞❛s ♣♦r ♣♦♥t❡s ❞❡ t❛❧ ♠♦❞♦ q✉❡ é ♣♦ssí✈❡❧ ❛♥❞❛r ❞❡ ✉♠❛ ✐❧❤❛ q✉❛❧q✉❡r ❛té q✉❛❧q✉❡r ♦✉tr❛✳ ❯♠ t✉r✐st❛ ♣❡r❝♦rr❡✉ t♦❞❛s ❛s ✐❧❤❛s ❝r✉③❛♥❞♦ ❝❛❞❛ ♣♦♥t❡ ❡①❛t❛♠❡♥t❡ ✉♠❛ ✈❡③✱ t❡♥❞♦ ✈✐s✐t❛❞♦ ❛ ✐❧❤❛ ❞❡ ❚r✐♣❧❛ três ✈❡③❡s✳ ◗✉❛♥t❛s ♣♦♥t❡s ❤á ❡♠ ❚r✐♣❧❛ s❡✿
✭❛✮♦ t✉r✐st❛ ♥ã♦ ❝♦♠❡ç♦✉ ♥❡♠ t❡r♠✐♥♦✉ s❡✉ ♣❡r❝✉rs♦ ❡♠ ❚r✐♣❧❛❀
❈❛♣ít✉❧♦ ✶✳ ❚❊❖❘■❆ ❉❖❙ ●❘❆❋❖❙ ✶✳✸✳ ■s♦♠♦r✜s♠♦
✭❜✮♦ t✉r✐st❛ ❝♦♠❡ç♦✉ s❡✉ ♣❡r❝✉rs♦ ❡♠ ❚r✐♣❧❛✱ ♠❛s ♥ã♦ t❡r♠✐♥♦✉ ❧á❀ ✭❝✮♦ t✉r✐st❛ ❝♦♠❡ç♦✉ ❡ t❡r♠✐♥♦✉ s❡✉ ♣❡r❝✉rs♦ ❡♠ ❚r✐♣❧❛✳
Pr✐♠❡✐r❛♠❡♥t❡✱ ❛♥❛❧✐s❡♠♦s ♦ ❡♥✉♥❝✐❛❞♦✱ ♦ ❣r✉♣♦ ❞❡ ✐❧❤❛s ❡stá ❧✐❣❛❞♦ ❞❡ ♠♦❞♦ q✉❡ é ♣♦ssí✈❡❧ ❛♥❞❛r ❞❡ ✉♠❛ ✐❧❤❛ q✉❛❧q✉❡r ❛té q✉❛❧q✉❡r ♦✉tr❛✱ ✐ss♦ ❞á ❛ ✐❞❡✐❛ ❞❡ r❡♣r❡s❡♥t❛çã♦ ♣♦r ✉♠ ❣r❛❢♦ ❝♦♥❡①♦❀ ♥♦ q✉❛❧✱ ❛s ✐❧❤❛s sã♦ ♦s ✈ért✐❝❡s ❡ ❛s ♣♦♥t❡s sã♦ ❛s ❛r❡st❛s✳ ❈♦♠♦ ♦ t✉r✐st❛ ❢♦✐ ❛ t♦❞❛s ❛s ✐❧❤❛s ❝r✉③❛♥❞♦ ❝❛❞❛ ♣♦♥t❡ ❡①❛t❛♠❡♥t❡ ✉♠❛ ✈❡③✱ t❡♠♦s ✉♠ ♣❛ss❡✐♦✳ ✭❛✮ ❙❡ ♦ t✉r✐st❛ ♥ã♦ ❝♦♠❡ç♦✉ ♥❡♠ t❡r♠✐♥♦✉ s❡✉ ♣❡r❝✉rs♦ ❡♠ ❚r✐♣❧❛✱ ❡ ♣❛ss♦✉ ♣♦r ❡❧❛ ✸ ✈❡③❡s✱ ♣r❡❝✐s♦✉ ❞❡ ✸ ♣♦♥t❡s ♣❛r❛ ❡♥tr❛r ❡ ✸ ♣♦♥t❡s ♣❛r❛ s❛✐r✳ P♦r ❝♦♥s❡❣✉✐♥t❡✱ ❤á ✻ ♣♦♥t❡s ❡♠ ❚r✐♣❧❛✳
✭❜✮ ❙❡ ❡❧❡ ❝♦♠❡ç♦✉ s❡✉ ♣❛ss❡✐♦ ♣♦r tr✐♣❧❛ ❡ ♥ã♦ t❡r♠✐♥♦✉ ❧á✱ t❡♠♦s ✉♠ ♣❛ss❡✐♦ ❛❜❡rt♦✱ q✉❡ ✐♥✐❝✐❛ ♥♦ ✈ért✐❝❡ ❡ ♥ã♦ ✈♦❧t❛ ❛ ❡❧❡✱ ❝♦♠♦ r❡s✉❧t❛❞♦ ❡ss❡ ✈ért✐❝❡ t❡♠ ❣r❛✉ í♠♣❛r✱ ❝♦♠♦ ❡❧❡ ✈♦❧t❛ ❛ ❚r✐♣❧❛ ✷ ✈❡③❡s ❡ s❛✐ ❞❡ ❧á✱ t❡♠♦s ✺ ❛r❡st❛s✱ ♦✉ s❡❥❛✱ ✺ ♣♦♥t❡s✳
✭❝✮ ◆❡ss❛ ❧❡tr❛ ❝✱ t❡♠♦s ✉♠❛ tr✐❧❤❛✱ ❡♠ ✈✐st❛ ❞✐ss♦✱ ♦ ❣r❛✉ ❞♦ ✈ért✐❝❡ q✉❡ r❡♣r❡s❡♥t❛ ❚r✐♣❧❛ é ♣❛r✳ P♦✐s ♦ ♣❛ss❡✐♦ ✐♥✐❝✐❛ ❡ t❡r♠✐♥❛ ♥♦ ♠❡s♠♦ ✈ért✐❝❡✳ ➱ ♥❡❝❡ssár✐♦ ✉♠❛ ❛r❡st❛ ♣❛r❛ s❛✐r ❞❡ ❚r✐♣❧❛✱ ♠❛✐s ✷ ♣❛r❛ ❡♥tr❛r ❡ s❛✐r ❞❡ ❚r✐♣❧❛ ♣❡❧❛ s❡❣✉♥❞❛ ✈❡③✱ ❡ ♠❛✐s ✶ ♣❛r❛ ✈♦❧t❛r✱ r❡s✉❧t❛♥❞♦ ❡♠ ✹ ❛r❡st❛s✳
✶✳✸ ■s♦♠♦r✜s♠♦
❖❜s❡r✈❡ ♦s ❣r❛❢♦s ❞❛ ✜❣✉r❛ ✶✳✶✵
❋✐❣✉r❛ ✶✳✶✵✿ ●r❛❢♦s ✐s♦♠♦r❢♦s
❊❧❡s sã♦ ♣❛r❡❝✐❞♦s❄ ❙❡r✐❛ ♣♦ssí✈❡❧ ❡st❛❜❡❧❡❝❡r ✉♠❛ r❡❧❛çã♦ ❞❡ ❜✐❥❡çã♦ ❡♥tr❡ s❡✉s ✈ért✐❝❡s❄ ❊❧❡s ♣♦❞❡r✐❛♠ r❡♣r❡s❡♥t❛r ✉♠❛ ♠❡s♠❛ s✐t✉❛çã♦❄
