INTRODUÇÃO À TEORIA DOS GRAFOS

❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞❛ P❛r❛í❜❛

❈❡♥tr♦ ❞❡ ❈✐ê♥❝✐❛s ❊①❛t❛s ❡ ❞❛ ◆❛t✉r❡③❛

❉❡♣❛rt❛♠❡♥t♦ ❞❡ ▼❛t❡♠át✐❝❛

▼❡str❛❞♦ Pr♦✜ss✐♦♥❛❧ ❡♠ ▼❛t❡♠át✐❝❛

❡♠ ❘❡❞❡ ◆❛❝✐♦♥❛❧ P❘❖❋▼❆❚

■♥tr♦❞✉çã♦ à ❚❡♦r✐❛ ❞♦s ●r❛❢♦s

♣♦r

●✐❧❞s♦♥ ❙♦❛r❡s ❞❡ ▼❡❧♦

s♦❜ ♦r✐❡♥t❛çã♦ ❞♦

Pr♦❢❡ss♦r ❉r✳ ◆❛♣♦❧❡ó♥ ❈❛r♦ ❚✉❡st❛

❉✐ss❡rt❛çã♦ ❛♣r❡s❡♥t❛❞❛ ❛♦ ❈♦r♣♦ ❉♦✲ ❝❡♥t❡ ❞♦ ▼❡str❛❞♦ Pr♦✜ss✐♦♥❛❧ ❡♠ ▼❛✲ t❡♠át✐❝❛ ❡♠ ❘❡❞❡ ◆❛❝✐♦♥❛❧ P❘❖❋▼❆❚✲ ❈❈❊◆✲❯❋P❇✱ ❝♦♠♦ r❡q✉✐s✐t♦ ♣❛r❝✐❛❧ ♣❛r❛ ♦❜t❡♥çã♦ ❞♦ tít✉❧♦ ❞❡ ▼❡str❡ ❡♠ ▼❛t❡♠át✐❝❛✳

■♥tr♦❞✉çã♦ à ❚❡♦r✐❛ ❞♦s ●r❛❢♦s

♣♦r

●✐❧❞s♦♥ ❙♦❛r❡s ❞❡ ▼❡❧♦

❉✐ss❡rt❛çã♦ ❛♣r❡s❡♥t❛❞❛ ❛♦ ❈♦r♣♦ ❉♦❝❡♥t❡ ❞♦ ▼❡str❛❞♦ Pr♦✜ss✐♦♥❛❧ ❡♠ ▼❛t❡♠át✐❝❛ ❡♠ ❘❡❞❡ ◆❛❝✐♦♥❛❧ P❘❖❋▼❆❚ ❈❈❊◆✲❯❋P❇✱ ❝♦♠♦ r❡q✉✐s✐t♦ ♣❛r❝✐❛❧ ♣❛r❛ ♦❜t❡♥çã♦ ❞♦ tít✉❧♦ ❞❡ ▼❡str❡ ❡♠ ▼❛t❡♠át✐❝❛✳

ár❡❛ ❞❡ ❈♦♥❝❡♥tr❛çã♦✿ ▼❛t❡♠át✐❝❛✳ ❆♣r♦✈❛❞❛ ♣♦r✿

Pr♦❢✳ ❉r✳ ◆❛♣♦❧❡ó♥ ❈❛r♦ ❚✉❡st❛ ✲❯❋P❇ ✭❖r✐❡♥t❛❞♦r✮

Pr♦❢✳ ❉r✳ ▲✐③❛♥❞r♦ ❙❛♥❝❤❡③ ❈❤❛❧❧❛♣❛ ✲ ❯❋P❇

Pr♦❢✳ ❉r✳ ▼✐❣✉❡❧ ❋✐❞❡♥❝✐♦ ▲♦❛②③❛ ▲♦③❛♥♦ ✲ ❯❋P❊

❆❣r❛❞❡❝✐♠❡♥t♦s

◗✉❡r♦ ❛❣r❛❞❡❝❡r ❛ ❉❡✉s ♣❡❧♦ ❞♦♠ ❞❛ ✈✐❞❛ ❡ ♣♦r ♠❡ ♣♦ss✐❜✐❧✐t❛r ❛❧❝❛♥ç❛r ♠❛✐s ❡st❛ ❝♦♥q✉✐st❛✳

➚ ♠✐♥❤❛ ❡s♣♦s❛ ❚❤❛❧✐t❛ ❘❛❢❛❡❧❛ ❞❛ ❙✐❧✈❛ ❇♦r❣❡s ❙♦❛r❡s ♣❡❧♦ ❛♠♦r ❡ ❛♣♦✐♦ ✐♥❝♦♥✲ ❞✐❝✐♦♥❛❧ ♥❡st❛ ❥♦r♥❛❞❛ ❞♦ ▼❡str❛❞♦ ❡ ♣♦r ❡st❛r ❛♦ ♠❡✉ ❧❛❞♦ ❡♠ t♦❞❛s ❛s ❞❡❝✐sõ❡s ❞❛ ♠✐♥❤❛ ✈✐❞❛✳

❆♦s ♠❡✉s ♣❛✐s ●✐❧❞♦ ❏♦sé ❞❡ ▼❡❧♦ ❡ ■✈❛♥✐❧③❛ ❙♦❛r❡s ❞❡ ▼❡❧♦ ♣♦r s❡r❡♠ ♠✐♥❤❛ ❜❛s❡ ❡ ♣♦r t♦❞♦ ❛♠♦r✱ ❝❛r✐♥❤♦ ❡ ❛♣♦✐♦ q✉❡ ♠❡ ❢❡③ s❡r ♦ ❤♦♠❡♠ q✉❡ s♦✉ ❤♦❥❡✳

➚ ♠✐♥❤❛ ✐r♠ã ●❛❜r✐❡❧❛ ❙♦❛r❡s ❞❡ ▼❡❧♦ ♣❡❧♦ s❡✉ ❛♠♦r ❢r❛t❡r♥❛❧✳

➚ ♠✐♥❤❛ s♦❜r✐♥❤❛ ❡ ❛✜❧❤❛❞❛ ▲❛r❛ ❙♦❛r❡s ❞❡ ▼❡❧♦ ❇❡③❡rr❛ ▲❡✐t❡ ♣♦r s✉❛ ❞♦ç✉r❛ ❞❡ ❝r✐❛♥ç❛ q✉❡ ♠❡ ❛❧❡❣r❛ t♦❞♦s ♦s ❞✐❛s✳

➚ ❈♦♦r❞❡♥❛çã♦ ❡ t♦❞♦s ♦s Pr♦❢❡ss♦r❡s ❞♦ P❘❖❋▼❆❚✳

❆♦ ❈♦♦r❞❡♥❛❞♦r ❞♦ P❘❖❋▼❆❚ Pr♦❢❡ss♦r ❉r✳ ❇r✉♥♦ ❍❡♥r✐q✉❡ ❈❛r✈❛❧❤♦ ❘✐❜❡✐r♦ ♣♦r t♦❞♦ s✉♣♦rt❡ ❡ ❛♣♦✐♦ ♥♦ ❞❡❝✉rs♦ ❞❡st❡ ▼❡str❛❞♦✳

❆♦s ♠❡♠❜r♦s ❞❛ ❇❛♥❝❛✱ ❡♠ ❡s♣❡❝✐❛❧ ❛♦ ♠❡✉ ❖r✐❡♥t❛❞♦r Pr♦❢❡ss♦r ❉r✳ ◆❛♣♦❧❡ó♥ ❈❛r♦ ❚✉❡st❛✱ ♣♦r t♦❞❛ ❛ ❢♦rç❛ ❞❛❞❛ ♥❡st❡s ❞♦✐s ❛♥♦s ❞❡ ▼❡str❛❞♦✱ ♣❡❧❛s ❝♦♥tr✐❜✉✐çõ❡s✱ s✉❣❡stõ❡s✱ ❝rít✐❝❛s q✉❡ ❝♦♥tr✐❜✉ír❛♠ ♣❛r❛ ❡❧❛❜♦r❛çã♦ ❞❡st❡ ❚r❛❜❛❧❤♦ ❞❡ ❈♦♥❝❧✉sã♦ ❞❡ ❈✉rs♦✳

❆ t♦❞♦s ♦s ❞❡♠❛✐s ❢❛♠✐❧✐❛r❡s ❡ ❛♠✐❣♦s ♣❡❧♦ ❛♣♦✐♦ ❡ t♦r❝✐❞❛ ❞❡ s❡♠♣r❡✳

❆♦s ♠❡✉s ❛♠✐❣♦s ❞❛ t✉r♠❛ ✷✵✶✷ ❞♦ P❘❖❋▼❆❚✱ ❡♠ ❡s♣❡❝✐❛❧ ❛♦s ❛♠✐❣♦s ❝♦♥✲ ❣r✉❡♥t❡s✿ ❆❧❡ss❛♥❞r♦ ▼✐❣♥❛❝✱ ❲❛s❤✐♥❣t♦♥✱ ▼❛❣♥✉♠✱ ❈②❜❡❧❡ ❡ ❆♥❞ré✱ ❡ ❛s ❛♠✐❣❛s ▼❛②❛♥❛ ❈②❜❡❧❡ ❡ ❊❞❥❛♥❡ ♣❡❧♦s ❡st✉❞♦s✱ ♣❡❧❛ ❛♠✐③❛❞❡ ❡ ♣❡❧❛ ❢♦rç❛ q✉❡ ♠❡ ❢♦✐ ❞❛❞❛ ♥❛ ❝❛♠✐♥❤❛❞❛ ❞♦ ▼❡str❛❞♦ P❘❖❋▼❆❚✳

❉❡❞✐❝❛tór✐❛

❘❡s✉♠♦

❊st❡ tr❛❜❛❧❤♦ ❛♣r❡s❡♥t❛ ✉♠ ❡st✉❞♦ ✐♥tr♦❞✉tór✐♦ s♦❜r❡ ❚❡♦r✐❛ ❞♦s ●r❛❢♦s✱ ❝♦♥s✐✲ ❞❡r❛♥❞♦ s✉❛ r❡❧❡✈â♥❝✐❛ ♣❛r❛ ♦ ❡♥s✐♥♦ ❞❛ ▼❛t❡♠át✐❝❛✳ ■♥✐❝✐❛❧♠❡♥t❡ é ❛♣r❡s❡♥t❛❞♦ ✉♠ ❜r❡✈❡ ❤✐stór✐❝♦ s♦❜r❡ ❛ ❚❡♦r✐❛ ❞♦s ●r❛❢♦s✳ ❊♠ s❡❣✉✐❞❛✱ ♦ ❝❛♣ít✉❧♦ ✶ é ❝♦♥st✐t✉í❞♦ ♣♦r ❛❧❣✉♠❛s ❞❡✜♥✐çõ❡s s♦❜r❡ ❣r❛❢♦s ❡ ❡①❡♠♣❧♦s✱ ♦ ❝❛♣ít✉❧♦ ✷ tr❛t❛ ❞♦s ❝❛♠✐♥❤♦s✱ ♣❛ss❡✐♦s ❡ ❝✐❝❧♦s ♥✉♠ ❣r❛❢♦✱ ❞❡st❛❝❛♥❞♦✲s❡ ♦s ♣❛ss❡✐♦s ❊✉❧❡r✐❛♥♦s ❡ ♦s ❝✐❝❧♦s ❍❛♠✐❧t♦♥✐❛♥♦s✱ ❛❜♦r❞❛♠♦s t❛♠❜é♠ ✉♠ t✐♣♦ ❡s♣❡❝✐❛❧ ❞❡ ❣r❛❢♦s✱ ❛s ár✈♦r❡s✳ ◆♦ ❝❛♣✐t✉❧♦ ✸ ❛❜♦r❞❛♠♦s ❛ ♣❧❛♥❛r✐❞❛❞❡ ♥♦s ❣r❛❢♦s✱ ❛♣r❡s❡♥t❛♥❞♦ ❛ss✐♠ ❛ ❋ór♠✉❧❛ ❞❡ ❊✉❧❡r✳ ❋✐♥❛❧♠❡♥t❡ ♥♦ ❝❛♣ít✉❧♦ ✹ ❛♣r❡s❡♥t❛♠♦s ❛❧❣✉♥s ♣r♦❜❧❡♠❛s ❡♥✈♦❧✈❡♥❞♦ ❣r❛❢♦s✳

❆❜str❛❝t

❚❤✐s ♣❛♣❡r ♣r❡s❡♥ts ❛♥ ✐♥tr♦❞✉❝t♦r② st✉❞② ♦❢ ❣r❛♣❤ t❤❡♦r②✱ ❝♦♥s✐❞❡r✐♥❣ ✐ts r❡❧❡✲ ✈❛♥❝❡ t♦ t❤❡ t❡❛❝❤✐♥❣ ♦❢ ♠❛t❤❡♠❛t✐❝s✳ ■♥✐t✐❛❧❧② ♣r❡s❡♥ts ❛ ❜r✐❡❢ ❤✐st♦r② ♦♥ ●r❛♣❤ ❚❤❡♦r②✳ ❚❤❡♥ t❤❡ ✜rst ❝❤❛♣t❡r ❝♦♥s✐sts ♦❢ s♦♠❡ ❞❡✜♥✐t✐♦♥s ♦♥ ❣r❛♣❤s ❛♥❞ ❡①❛♠♣❧❡s✱ t❤❡ s❡❝♦♥❞ ❝❤❛♣t❡r ❞❡❛❧s ✇✐t❤ t❤❡ ♣❛t❤s✱ ✇❛❧❦s ❛♥❞ ❝②❝❧❡s ✐♥ ❛ ❣r❛♣❤✱ ❤✐❣❤❧✐❣❤t✐♥❣ t❤❡ ❊✉❧❡r✐❛♥ t♦✉rs ❛♥❞ ❍❛♠✐❧t♦♥✐❛♥ ❝②❝❧❡s✱ ❛❧s♦ ❜♦❛r❞❡❞ ❛ s♣❡❝✐❛❧ t②♣❡ ♦❢ ❣r❛♣❤s✱ tr❡❡s ✳ ■♥ ❈❤❛♣t❡r ✸ ✇❡ ❛❞❞r❡ss t❤❡ ♣❧❛♥❛r✐t② ✐♥ ❣r❛♣❤s✱ t❤✉s ♣r❡s❡♥t✐♥❣ ❊✉❧❡r✬s ❋♦r♠✉❧❛✳ ❋✐♥❛❧❧② ✐♥ ❈❤❛♣t❡r ✹ ✇❡ ♣r❡s❡♥t s♦♠❡ ♣r♦❜❧❡♠s ✐♥✈♦❧✈✐♥❣ ❣r❛♣❤s✳

❙✉♠ár✐♦

✶ ●r❛❢♦s ✶

✶✳✶ ❉❡✜♥✐çõ❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶ ✶✳✷ ❆❧❣✉♥s ❊①❡♠♣❧♦s ❞❡ ●r❛❢♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻

✷ P❛ss❡✐♦s✱ ❈❛♠✐♥❤♦s✱ ❈✐❝❧♦s ❡ ➪r✈♦r❡s ✽

✷✳✶ ■❞❡✐❛s ❇ás✐❝❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽ ✷✳✷ P❛ss❡✐♦s ❊✉❧❡r✐❛♥♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✶ ✷✳✸ ❈✐❝❧♦s ❍❛♠✐❧t♦♥✐❛♥♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✷ ✷✳✹ ➪r✈♦r❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✸

✸ P❧❛♥❛r✐❞❛❞❡ ✶✻

✸✳✶ ❘❡♣r❡s❡♥t❛çõ❡s ❡ ❈r✉③❛♠❡♥t♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✻ ✸✳✷ ❋ór♠✉❧❛ ❞❡ ❊✉❧❡r ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✽

✹ ❆❧❣✉♥s Pr♦❜❧❡♠❛s ❊♥✈♦❧✈❡♥❞♦ ●r❛❢♦s ✷✵

✹✳✶ ❖ Pr♦❜❧❡♠❛ ❞❛s P♦♥t❡s ❞❡ ❑ö♥✐❣s❜❡r❣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✵ ✹✳✷ ❖ Pr♦❜❧❡♠❛ ❈❤✐♥ês ❞♦ ❈❛rt❡✐r♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✷ ✹✳✸ ❖ Pr♦❜❧❡♠❛ ❞❛s ✹ ❈♦r❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✸

❘❡❢❡rê♥❝✐❛s ❇✐❜❧✐♦❣rá✜❝❛s ✷✺

■♥tr♦❞✉çã♦

❆ ❚❡♦r✐❛ ❞♦s ●r❛❢♦s t❡♠ ✉♠❛ ♦r✐❣❡♠ r❡❧❛t✐✈❛♠❡♥t❡ r❡❝❡♥t❡ ✭sé❝✉❧♦ ❳❱■■■✮ ♥❛ ❤✐stór✐❛ ❞❛ ▼❛t❡♠át✐❝❛✳ ❉❡s❡♥✈♦❧✈✐❞❛ ❥á ♥♦ sé❝✉❧♦ ❳❳✱ ❝✉❥❛ ✐♠♣♦rtâ♥❝✐❛ s❡ t❡♠ ✐♠♣♦st♦ ♣♦r s✉❛s ❧✐❣❛çõ❡s ❡ ❛♣❧✐❝❛çõ❡s ❡♠ ♦✉tr❛s ❝✐ê♥❝✐❛s✱ ❜❡♠ ❝♦♠♦ ❡♠ ♦✉tr❛s ár❡❛s ❞❛ ▼❛t❡♠át✐❝❛✳

■♥❢♦r♠❛❧♠❡♥t❡✱ ✉♠ ❣r❛❢♦ ♣♦❞❡ s❡r ✈✐st♦ ❝♦♠♦ ✉♠ ❝♦♥❥✉♥t♦ ❞❡ ♣♦♥t♦s✱ ❝❤❛♠❛❞♦s ✈ért✐❝❡s✱ ❡ ♦✉tr♦ ❞❡ ♣❛r❡s ❞❡ss❡s ♣♦♥t♦s✱ ❝❤❛♠❛❞♦s ❛r❡st❛s❀ ❝❛❞❛ ❛r❡st❛ ❧✐❣❛ ✉♠ ♣❛r ❞❡ ♣♦♥t♦s ✭❡①tr❡♠✐❞❛❞❡s✮ q✉❡ ❛ ❞❡t❡r♠✐♥❛✳ ❆ r❡♣r❡s❡♥t❛çã♦ ✉s✉❛❧ é ❢❡✐t❛ ♣♦r ❧✐♥❤❛s ✭❛r❡st❛s✮ ❛ ❧✐❣❛r ♣♦♥t♦s ❞♦ ♣❧❛♥♦ ✭✈ért✐❝❡s✮✳

❋✐❣✉r❛ ✶✿ ❘❡♣r❡s❡♥t❛çã♦ ❞❡ ✉♠ ❣r❛❢♦

❚❛❧ ❝♦♥❝❡✐t♦ s✐♠♣❧❡s t♦r♥❛ ❝❧❛r♦ q✉❡ ❡❧❡ ♣❡r♠✐t❡ ❛ ♠♦❞❡❧❛❣❡♠ ❞❡ s✐t✉❛çõ❡s ❝♦♥✲ ❝r❡t❛s ❝♦♠♦✿ r❡❞❡s ❞❡ ❝♦♠♣✉t❛❞♦r❡s✱ ❞❡ ❝♦♠✉♥✐❝❛çõ❡s✱ ❛ ❲❡❜ ✭❧✐❣❛çã♦ ❢ís✐❝❛ ❡♥tr❡ ♦s ♥ós ❞❛ r❡❞❡✮✱ ár✈♦r❡s ❣❡♥❡❛❧ó❣✐❝❛s✱ ◗✉í♠✐❝❛ ❖r❣â♥✐❝❛ ✭✐sô♠❡r♦s✮ ❡t❝✳

❈♦♥t✉❞♦✱ ❛ ❧✐❣❛ç❛♦ ❢ís✐❝❛ ♥ã♦ é ♥❡❝❡ssár✐❛❀ t❛♠❜é♠ ♣♦❞❡ s❡r ❛ss♦❝✐❛❞♦ ✉♠ ❣r❛❢♦ ❛ ✉♠ q✉❛❧q✉❡r ❝♦♥❥✉♥t♦ ♥♦ q✉❛❧ ❡st❡❥❛ ❞❡✜♥✐❞❛ ✉♠❛ r❡❧❛çã♦ ❜✐♥ár✐❛✱ ❝♦♠♦ ❛ r❡❧❛çã♦ ✭❛ é ♣r✐♠♦ ❝♦♠ ❜✮ q✉❡ ❞❡t❡r♠✐♥❛ ✉♠ ❣r❛❢♦ ♥✉♠ ❝♦♥❥✉♥t♦ ✜①❛❞♦ ❞❡ ✐♥t❡✐r♦s✱ ♦✉ ❛ r❡❧❛çã♦ ✭❛ é ✜❧❤♦ ❞❡ ❜✮ q✉❡ ♣❡r♠✐t❡ ❛ss♦❝✐❛r ❛ ✉♠❛ ❞❛❞❛ ❢❛♠í❧✐❛ ✉♠ ❣r❛❢♦ ✭ár✈♦r❡ ❣❡♥❡❛❧ó❣✐❝❛✮✳

❖ t❡r♠♦ ❣r❛❢♦ ❢♦✐ ✉s❛❞♦ ♣❡❧❛ ♣r✐♠❡✐r❛ ✈❡③ ♣♦r ❏❛♠❡s ❏♦s❡♣❤ ❙②❧✈❡st❡r ♥✉♠ ❛rt✐❣♦ ♣✉❜❧✐❝❛❞♦ ❡♠ ✶✽✼✼✱ ♥❛ ◆❛t✉r❡✳

❋✐❣✉r❛ ✷✿ ❏❛♠❡s ❏♦s❡♣❤ ❙②❧✈❡st❡r

◆♦ ❡♥t❛♥t♦✱ ❛♣❡s❛r ❞❡st❛ ♣r✐♠❡✐r❛ ❞❛t❛ ❞❡ r❡❢❡rê♥❝✐❛ ❛♦ t❡r♠♦ ❣r❛❢♦ ❡ ❞❡ s✉❛ ❞❡✜♥✐çã♦ ❢♦r♠❛❧ só s✉r❣✐r ♥♦ sé❝✉❧♦ ❳❳✱ ❛ r❡s♦❧✉çã♦ ❞❡ ❊✉❧❡r ❞♦ Pr♦❜❧❡♠❛ ❞❛s P♦♥t❡s ❞❡ ❑ö♥✐❣s❜❡r❣✱ ♣✉❜❧✐❝❛❞❛ ❡♠ ✶✼✸✻✱ é r❡❢❡r✐❞❛ ❝♦♠♦ ❛ ♣r✐♠❡✐r❛ ♣✉❜❧✐❝❛çã♦ ❞❛ t❡♦r✐❛✳

❋✐❣✉r❛ ✸✿ ▲❡♦♥❤❛r❞ ❊✉❧❡r

❖ ♣r♦❜❧❡♠❛ é ❜❛s❡❛❞♦ ♥❛ ❝✐❞❛❞❡ ❞❡ ❑ö♥✐❣s❜❡r❣ ✭t❡rr✐tór✐♦ ❞❛ Prúss✐❛ ❛té ✶✾✹✺✱ ❛t✉❛❧ ❑❛❧✐♥✐♥❣r❛❞♦✮✱ q✉❡ é ❝♦rt❛❞❛ ♣❡❧♦ ❘✐♦ Pr❡❣ó❧✐❛✱ ♦♥❞❡ ❤á ❞✉❛s ❣r❛♥❞❡s ✐❧❤❛s q✉❡✱ ❥✉♥t❛s✱ ❢♦r♠❛♠ ✉♠ ❝♦♠♣❧❡①♦ q✉❡ ♥❛ é♣♦❝❛ ❝♦♥t✐♥❤❛ s❡t❡ ♣♦♥t❡s✱ ❝♦♥❢♦r♠❡ ♠♦str❛ ❛ ✜❣✉r❛ ❛❜❛✐①♦✳ ❉❛s s❡t❡ ♣♦♥t❡s ♦r✐❣✐♥❛✐s✱ ✉♠❛ ❢♦✐ ❞❡♠♦❧✐❞❛ ❡ r❡❝♦♥str✉í❞❛ ❡♠ ✶✾✸✺✱ ❞✉❛s ❢♦r❛♠ ❞❡str✉í❞❛s ❞✉r❛♥t❡ ❛ ❙❡❣✉♥❞❛ ●✉❡rr❛ ▼✉♥❞✐❛❧ ❡ ♦✉tr❛s ❞✉❛s ❢♦r❛♠ ❞❡♠♦❧✐❞❛s ♣❛r❛ ❞❛r ❧✉❣❛r ❛ ✉♠❛ ú♥✐❝❛ ✈✐❛ ❡①♣r❡ss❛✳ ❆t✉❛❧♠❡♥t❡ ❛♣❡♥❛s ❞✉❛s ♣♦♥t❡s sã♦ ❞❛ é♣♦❝❛ ❞❡ ▲❡♦♥❤❛r❞ ❊✉❧❡r✳

❋✐❣✉r❛ ✹✿ ❆s P♦♥t❡s ❞❡ ❑ö♥✐❣s❜❡r❣ ❡ s✉❛ r❡♣r❡s❡♥t❛çã♦ ❡♠ ❣r❛❢♦

❉✐s❝✉t✐❛✲s❡ ♥❛s r✉❛s ❞❛ ❝✐❞❛❞❡ ❛ ♣♦ss✐❜✐❧✐❞❛❞❡ ❞❡ ❛tr❛✈❡ss❛r t♦❞❛s ❛s ♣♦♥t❡s s❡♠ r❡♣❡t✐r ♥❡♥❤✉♠❛✳ ❍❛✈✐❛✲s❡ t♦r♥❛❞♦ ✉♠❛ ❧❡♥❞❛ ♣♦♣✉❧❛r ❛ ♣♦ss✐❜✐❧✐❞❛❞❡ ❞❛ ❢❛ç❛♥❤❛ q✉❛♥❞♦ ❊✉❧❡r✱ ❡♠ ✶✼✸✻✱ ♣r♦✈♦✉ q✉❡ ♥ã♦ ❡①✐st✐❛ ❝❛♠✐♥❤♦ q✉❡ ♣♦ss✐❜✐❧✐t❛ss❡ t❛✐s r❡s✲ tr✐çõ❡s✳

❊✉❧❡r ✉s♦✉ ✉♠ r❛❝✐♦❝í♥✐♦ ♠✉✐t♦ s✐♠♣❧❡s✳ ❚r❛♥s❢♦r♠♦✉ ♦s ❝❛♠✐♥❤♦s ❡♠ r❡t❛s ❡ s✉❛s ✐♥t❡rs❡❝çõ❡s ❡♠ ♣♦♥t♦s✱ ❝r✐❛♥❞♦ ♣♦ss✐✈❡❧♠❡♥t❡ ♦ ♣r✐♠❡✐r♦ ❣r❛❢♦ ❞❛ ❤✐stór✐❛✳ ❊♥tã♦ ♣❡r❝❡❜❡✉ q✉❡ só s❡r✐❛ ♣♦ssí✈❡❧ ❛tr❛✈❡ss❛r ♦ ❝❛♠✐♥❤♦ ✐♥t❡✐r♦ ♣❛ss❛♥❞♦ ✉♠❛ ú♥✐❝❛ ✈❡③ ❡♠ ❝❛❞❛ ♣♦♥t❡ s❡ ❤♦✉✈❡ss❡ ❡①❛t❛♠❡♥t❡ ③❡r♦ ♦✉ ❞♦✐s ♣♦♥t♦s ❞❡ ♦♥❞❡ s❛íss❡ ✉♠ ♥ú♠❡r♦ í♠♣❛r ❞❡ ❝❛♠✐♥❤♦s✳ ❆ r❛③ã♦ ❞❡ t❛❧ ❝♦✐s❛ é q✉❡ ❞❡ ❝❛❞❛ ♣♦♥t♦ ❞❡✈❡ ❤❛✈❡r ✉♠ ♥ú♠❡r♦ ♣❛r ❞❡ ❝❛♠✐♥❤♦s✱ ♣♦✐s s❡rá ♣r❡❝✐s♦ ✉♠ ❝❛♠✐♥❤♦ ♣❛r❛ ✧❡♥tr❛r✧❡ ♦✉tr♦ ♣❛r❛ ✧s❛✐r✧✳ ❖s ❞♦✐s ♣♦♥t♦s ❝♦♠ ❝❛♠✐♥❤♦s í♠♣❛r❡s r❡❢❡r❡♠✲s❡ ❛♦ ✐♥í❝✐♦ ❡ ❛♦ ✜♥❛❧ ❞♦ ♣❡r❝✉rs♦✱ ♣♦✐s ❡st❡s ♥ã♦ ♣r❡❝✐s❛♠ ❞❡ ✉♠ ♣❛r❛ ❡♥tr❛r ❡ ✉♠ ♣❛r❛ s❛✐r✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✳ ❙❡ ♥ã♦ ❤♦✉✈❡r ♣♦♥t♦s ❝♦♠ ♥ú♠❡r♦ í♠♣❛r ❞❡ ❝❛♠✐♥❤♦s✱ ♣♦❞❡✲s❡ ✭❡ ❞❡✈❡✲s❡✮ ✐♥✐❝✐❛r ❡ t❡r♠✐♥❛r ♦ tr❛❥❡t♦ ♥♦ ♠❡s♠♦ ♣♦♥t♦✱ ♣♦❞❡♥❞♦ ❡ss❡ s❡r q✉❛❧q✉❡r ♣♦♥t♦ ❞♦ ❣r❛❢♦✳ ■ss♦ ♥ã♦ é ♣♦ssí✈❡❧ q✉❛♥❞♦ t❡♠♦s ❞♦✐s ♣♦♥t♦s ❝♦♠ ♥ú♠❡r♦s í♠♣❛r❡s ❞❡ ❝❛♠✐♥❤♦s✱ s❡♥❞♦ ♦❜r✐❣❛t♦r✐❛♠❡♥t❡ ✉♠ ♦ ✐♥í❝✐♦ ❡ ♦✉tr♦ ♦ ✜♠✳

❉✉❛s ❞❛s s❡t❡ ♣♦♥t❡s ♦r✐❣✐♥❛✐s ❞❛ ❝✐❞❛❞❡ ❢♦r❛♠ ❞❡str✉í❞❛s ❞✉r❛♥t❡ ❞♦ ❜♦♠❜❛r✲ ❞❡❛♠❡♥t♦ ❞❡ ❑ö♥✐❣s❜❡r❣ ❡♠ ❛❣♦st♦ ❞❡ ✶✾✹✹✳

❈❛♣ít✉❧♦ ✶

●r❛❢♦s

✶✳✶ ❉❡✜♥✐çõ❡s

❊st❛ s❡çã♦ ❢♦✐ ❡❧❛❜♦r❛❞❛ ❛ ♣❛rt✐r ❞❛s s❡❣✉✐♥t❡s r❡❢❡rê♥❝✐❛s ❜✐❜❧✐♦❣rá✜✲ ❝❛s✿ ❬✶❪✱ ❬✷❪ ❡ ❬✸❪✳

❉❡✜♥✐çã♦ ✶✳✶ ❯♠ ❣r❛❢♦ ✭s✐♠♣❧❡s✮ G ❝♦♥s✐st❡ ❞❡ ✉♠ ❝♦♥❥✉♥t♦ ✜♥✐t♦ ❡ ♥ã♦ ✈❛③✐♦ V(G) ❞❡ ♦❜❥❡t♦s ❝❤❛♠❛❞♦s ✈ért✐❝❡s✱ ❥✉♥t❛♠❡♥t❡ ❝♦♠ ✉♠ ❝♦♥❥✉♥t♦ E(G) ❞❡ ♣❛r❡s

♥ã♦ ♦r❞❡♥❛❞♦s ❞❡ ✈ért✐❝❡s❀ ♦s ❡❧❡♠❡♥t♦s ❞❡ E(G) sã♦ ❝❤❛♠❛❞♦s ❞❡ ❛r❡st❛s✳ P♦❞❡✲

♠♦s r❡♣r❡s❡♥tá✲❧♦ ♣♦r G= (V;E)✱ ♦♥❞❡ V =V(G) ❡ E =E(G)✳

❙❡ G = (V;E) é ✉♠ ❣r❛❢♦ ❡ ✉ ❡ ✈ sã♦ ❞♦✐s ❞❡ s❡✉s ✈ért✐❝❡s✱ ❞✐r❡♠♦s q✉❡ ✉ ❡ ✈

sã♦ ❛❞❥❛❝❡♥t❡s s❡ {u,v} ∈ E❀ ♥❡st❡ ❝❛s♦✱ ❞✐③❡♠♦s ❛✐♥❞❛ q✉❡ ❛ ❛r❡st❛ {u,v} ✐♥❝✐❞❡

♥♦s ✈ért✐❝❡s ✉ ❡ ✈✳ P♦❞❡♠♦s ❞❡♥♦t❛r ❛ ❛r❡st❛ {u,v} s✐♠♣❧❡s♠❡♥t❡ ♣♦r ✉✈✱ s❡♠♣r❡

q✉❡ ♥ã♦ ❤♦✉✈❡r ♣❡r✐❣♦ ❞❡ ❝♦♥❢✉sã♦✳ ❙❡ ✉ ❡ ✈ ♥ã♦ ❢♦r❡♠ ❛❞❥❛❝❡♥t❡s✱ ❞✐r❡♠♦s q✉❡ sã♦ ✈ért✐❝❡s ♥ã♦ ❛❞❥❛❝❡♥t❡s ❞❡ G✳

●r❛❢♦s sã♦ ❣❡r❛❧♠❡♥t❡ r❡♣r❡s❡♥t❛❞♦s ♣♦r ❞✐❛❣r❛♠❛s✱ ♦♥❞❡ ♦s ❡❧❡♠❡♥t♦s ❞❡ V

❝♦rr❡s♣♦♥❞❡♠ ❛ ♣♦♥t♦s ♥♦ ♣❧❛♥♦ ❡ ❛s ❛r❡st❛s ❞❡ G❝♦rr❡s♣♦♥❞❡♠ ❛ ❛r❝♦s ❧✐❣❛♥❞♦ ♦s

✈ért✐❝❡s ❝♦rr❡s♣♦♥❞❡♥t❡s✳ ❆ ✜❣✉r❛ ❛ss✐♠ ♦❜t✐❞❛ ♥ã♦ t❡♠ ♥❡♥❤✉♠ s✐❣♥✐✜❝❛❞♦ ❣❡♦♠é✲ tr✐❝♦✱ s❡✉ ♣r♦♣ós✐t♦ s❡♥❞♦ s♦♠❡♥t❡ ♦ ❞❡ r❡♣r❡s❡♥t❛r ❡sq✉❡♠❛t✐❝❛♠❡♥t❡ ❛s r❡❧❛ç♦❡s ❞❡ ❛❞❥❛❝ê♥❝✐❛ ❡♥tr❡ ♦s ✈ért✐❝❡s ❞❡ G✳ P♦r ❡①❡♠♣❧♦✱ s❡

G= ({a, b, c, d};{{a, b},{a, c},{a, d},{b, c}})✱

❡♥tã♦G♣♦❞❡ s❡r r❡♣r❡s❡♥t❛❞♦ ♣♦r q✉❛❧q✉❡r ✉♠ ❞♦s ❞✐❛❣r❛♠❛s ❛❜❛✐①♦✱ ❤❛❥❛ ✈✐st❛

q✉❡ ❛♠❜♦s ❡♥❝❡rr❛♠ ❛s ♠❡s♠❛s r❡❧❛çõ❡s ❞❡ ❛❞❥❛❝ê♥❝✐❛✳

●r❛❢♦s ❈❛♣ít✉❧♦ ✶

❋✐❣✉r❛ ✶✳✶✿ r❡♣r❡s❡♥t❛çõ❡s ❞❡ ✉♠ ❣r❛❢♦✳

❉❡✜♥✐çã♦ ✶✳✷ ❯♠ ❣r❛✈♦ tr✐✈✐❛❧ é ✉♠ ❣r❛❢♦ ♥♦ q✉❛❧ G = (V;∅)✱ ♦✉ s❡❥❛✱ é ✉♠

❣r❛❢♦ q✉❡ ♣♦ss✉✐ n ✈ért✐❝❡s✱ ♣♦❞❡♥❞♦ s❡r r❡♣r❡s❡♥t❛❞♦ ♣♦r ❝♦♥❥✉♥t♦ ❞❡ n ♣♦♥t♦s ♥♦

♣❧❛♥♦✱ s❡♠ q✉❛✐sq✉❡r ❛r❝♦s ✭❛r❡st❛s✮✳

❉❡✜♥✐çã♦ ✶✳✸ ❯♠ ❣r❛❢♦ ❝♦♠♣❧❡t♦ é ✉♠ ❣r❛❢♦ q✉❡ ♣♦ss✉✐ n ✈ért✐❝❡s✱ ❝♦♠ ❞♦✐s

q✉❛✐sq✉❡r ❞❡❧❡s ❝♦♥❡❝t❛❞♦s ♣♦r ✉♠ ❛r❝♦✳ ❯♠ ❣r❛❢♦ ❝♦♠♣❧❡t♦ ❝♦♠ n ✈ért✐❝❡s é ❞❡♥♦✲

t❛❞♦ ♣♦r Kn❀ ❡♠ ♣❛rt✐❝✉❧❛r✱ Kn t❡♠ ❡①❛t❛♠❡♥t❡ n₂

❛r❡st❛s✳

❉❡✜♥✐çã♦ ✶✳✹ ❋✐①❛❞♦ ✉♠ ✈ért✐❝❡ u ❞❡ G✱ ❞❡♥♦t❛♠♦s ♣♦r NG(u) ♦ ❝♦♥❥✉♥t♦ ❞♦s

✈ért✐❝❡s ❛❞❥❛❝❡♥t❡s ✭♦✉ ✈✐③✐♥❤♦s✮ ❛ u✿ NG(u) ={v ∈V;uv ∈E}.

❉❡✜♥✐çã♦ ✶✳✺ ❙❡❥❛ G= (V, E) ✉♠ ❣r❛❢♦✳ ❙❡ u∈V✱ ♦ ❣r❛✉ ❞❡u✱ ❞❡♥♦t❛❞♦dG(u)✱

é ♦ ♥ú♠❡r♦ ❞❡ ✈ért✐❝❡s ❛❞❥❛❝❡♥t❡s ❛ u✿

d_G(u) = #N_G(u)✳

❚❡♦r❡♠❛ ✶✳✶ ✭❊✉❧❡r✮ ❊♠ ✉♠ ❣r❛❢♦ G= (V;E)✱ ❛ s♦♠❛ ❞♦s ❣r❛✉s ❞♦s ✈ért✐❝❡s é

s❡♠♣r❡ ✐❣✉❛❧ ❛♦ ❞♦❜r♦ ❞♦ ♥ú♠❡r♦ ❞❡ ❛r❡st❛s✳ ❊♠ sí♠❜♦❧♦s✿

2|E|=X

u∈V

dG(u). (1.1)

Pr♦✈❛✳

❯t✐❧✐③❡♠♦s ❝♦♥t❛❣❡♠ ❞✉♣❧❛✳ ❇❛st❛ ♦❜s❡r✈❛r s❡ ǫ = {u, v} é ✉♠❛ ❛r❡st❛ ❞❡ G✱

❡♥tã♦ ǫé ❝♦♥t❛❞❛ ❡①❛t❛♠❡♥t❡ ❞✉❛s ✈❡③❡s ♥♦ s❡❣✉♥❞♦ ♠❡♠❜r♦ ❞❡ ✭✶✳✶✮✿ ✉♠❛ ✈❡③ ♥❛

♣❛r❝❡❧❛dG(u)❡ ♦✉tr❛ ♥❛ ♣❛r❝❡❧❛ ❞❡ dG(v)✳ P♦rt❛♥t♦✱ ♦ s❡❣✉♥❞♦ ♠❡♠❜r♦ t❡♠ ❞❡ s❡r

✐❣✉❛❧ ❛ 2|E|✱ ❥á q✉❡ t❛❧ ♥ú♠❡r♦ t❛♠❜é♠ ❝♦♥t❛ ❝❛❞❛ ❛r❡st❛ ❡①❛t❛♠❡♥t❡ ❞✉❛s ✈❡③❡s✳

●r❛❢♦s ❈❛♣ít✉❧♦ ✶

❈♦r♦❧ár✐♦ ✶✳✶ ✭❊✉❧❡r✮ ❊♠ ✉♠ ❣r❛❢♦ q✉❛❧q✉❡r✱ ♦ ♥ú♠❡r♦ ❞❡ ✈ért✐❝❡s ❞❡ ❣r❛✉ í♠♣❛r é ♣❛r✳

Pr♦✈❛✳

❈♦♥s✐❞❡r❡ ♦ ❣r❛❢♦ G= (V, E)✳ ❉❡♥♦t❛♥❞♦ ♣♦r di ♦ ❣r❛✉ ❞♦ ✈ért✐❝❡ i✱ t❡♠♦s✱ ❞♦

❚❡♦r❡♠❛ ❞❡ ❊✉❧❡r✿

2|E| = X

i∈N

di =

X

i∈N|di par

di +

X

i∈N|di mpar

di

❙❡♣❛r❛♠♦s ♦ s♦♠❛tór✐♦ ❡♠ ❞✉❛s ♣❛r❝❡❧❛s✱ ❛ ♣r✐♠❡✐r❛ ❝♦♥t❡♥❞♦ ♦s ❣r❛✉s ♣❛r❡s ❡ ❛ s❡❣✉♥❞❛ ♦s í♠♣❛r❡s✳ ❊♥tã♦✱ ❛ ♣r✐♠❡✐r❛ ♣❛r❝❡❧❛ é ♣❛r✱ ♠❛s ❝♦♠♦ ❛ s♦♠❛ ❞❛s ❞✉❛s ♣❛r❝❡❧❛s t❛♠❜é♠ é ♣❛r ✭♦ ❞♦❜r♦ ❞❛ ❝❛r❞✐♥❛❧✐❞❛❞❡ ❞❡ E✮✱ ❛ s❡❣✉♥❞❛ ♣❛r❝❡❧❛ t❛♠❜é♠

é ♣❛r✳ ❖❜s❡r✈❡ ❛❣♦r❛ q✉❡✱ ♣❛r❛ t❡r ✉♠❛ s♦♠❛ ❞❡ ♣❛r❝❡❧❛s í♠♣❛r❡s r❡s✉❧t❛♥❞♦ ❡♠ ✉♠ ♥ú♠❡r♦ ♣❛r✱ ❞❡✈❡♠♦s t❡r ✉♠ ♥ú♠❡r♦ ♣❛r ❞❡ ♣❛r❝❡❧❛s✱ ♦ q✉❡ ❝♦♥❝❧✉✐ ❛ ❞❡♠♦♥str❛çã♦✳

❉❡✜♥✐çã♦ ✶✳✻ ❉♦✐s ❣r❛❢♦s G = (V1;E1) ❡ H = (V2;E2) sã♦ ✐s♦♠♦r❢♦s s❡ ❡①✐st✐r

✉♠❛ ❜✐❥❡çã♦ f :V1 →V2 q✉❡ ♣r❡s❡r✈❛ ✐♥❝✐❞ê♥❝✐❛✱ ✐✳❡✳✱ t❛❧ q✉❡✱ ♣❛r❛ ✈ért✐❝❡s ❞✐st✐♥t♦s

q✉❛✐sq✉❡r ✉ ❡ ✈ ❞❡ ●✱ t❡♥❤❛♠♦s

{u, v} ∈E₁ ⇔ {f(u), f(v)} ∈E₂✳

◆❡ss❡ ❝❛s♦✱ ❞❡♥♦t❛♠♦s G₁ ≃G₂✳

Pr♦♣♦s✐çã♦ ✶✳✶ ❉♦✐s ❣r❛❢♦s ✐s♦♠♦r❢♦s tê♠ q✉❛♥t✐❞❛❞❡ ✐❣✉❛✐s ❞❡ ✈ért✐❝❡s ❞❡ ✉♠ ♠❡s♠♦ ❣r❛✉✳ ❊♠ ♣❛rt✐❝✉❧❛r✱ tê♠ q✉❛♥t✐❞❛❞❡s ✐❣✉❛✐s ❞❡ ✈ért✐❝❡s ❡ q✉❛♥t✐❞❛❞❡s ✐❣✉❛✐s ❞❡ ❛r❡st❛s✳

Pr♦✈❛✳

❙❡❥❛♠ G = (V₁;E₁) ❡ H = (V₂;E₂) ❣r❛❢♦s ✐s♦♠♦r❢♦s ❡ f : V₁ → V₂ ✉♠❛ ❜✐❥❡çã♦

q✉❡ ♣r❡s❡r✈❛ ✐♥❝✐❞ê♥❝✐❛✳ ❙❡ué ✉♠ ✈ért✐❝❡ ❞❡G❝♦♠ ❣r❛✉k ≥0✱ é ✐♠❡❞✐❛t♦ q✉❡f(u)

é ✉♠ ✈ért✐❝❡ ❞❡H ❝♦♠ ❣r❛✉k✱ ❡ r❡❝✐♣r♦❝❛♠❡♥t❡ ✳ P♦rt❛♥t♦✱ G❡H tê♠ q✉❛♥t✐❞❛❞❡s

✐❣✉❛✐s ❞❡ ✈ért✐❝❡s ❞❡ ❣r❛✉ k✱ ❞✐❣❛♠♦s ak✱ ❞❡ ♠❛♥❡✐r❛ q✉❡

|V₁|=X

k≥0

ak=|V2|✳

P♦r ♦✉tr♦ ❧❛❞♦✱ ❛♣❧✐❝❛♥❞♦ ♦ t❡♦r❡♠❛ ❞❡ ❊✉❧❡r ❞✉❛s ✈❡③❡s✱ ❥✉♥t❛♠❡♥t❡ ❝♦♠ ❛ ♣r✐♠❡✐r❛ ♣❛rt❡ ❛❝✐♠❛✱ ♦❜t❡♠♦s

2|E₁|= X

u∈V1

dG(u) =

X

u∈V1

dH(f(u)) =

X

f(u)∈V₂

dH(f(u)) = 2|E2|

●r❛❢♦s ❈❛♣ít✉❧♦ ✶

❡✱ ❞❛í|E1|=|E2|✳

❊①❡♠♣❧♦ ✶✳✶ ❙❡ G = (V;E) ❢♦r ✉♠ ❣r❛❢♦ ❞❡ n ✈ért✐❝❡s✱ ♣♦❞❡♠♦s✱ s❡♠♣r❡ q✉❡

♥❡❝❡ssár✐♦✱ s✉♣♦r q✉❡ V = In✳ ❉❡ ❢❛t♦✱ ❝♦♠♦ |V| = n✱ ♣♦❞❡♠♦s ❡s❝♦❧❤❡r ✉♠❛

❜✐❥❡çã♦ f :V →In ❡ ❞❡✜♥✐r H = (In;F) ♣♦♥❞♦✱ ♣❛r❛ i6=j ❡♠ In✱

{i, j} ∈F ⇔ {f−1(i), f−1(j)} ∈E.

❉❡✜♥✐çã♦ ✶✳✼ ❉❛❞♦ ✉♠ ❣r❛❢♦ G = (V;E)✱ ❝♦♠ |V| = n✱ s✉♣♦♥❤❛✱ ❞❡ ❛❝♦r❞♦

❝♦♠ ♦ ❡①❡♠♣❧♦ ❛❝✐♠❛✱ q✉❡ V = In✳ ❆ ♠❛tr✐③ ❞❡ ❛❞❥❛❝ê♥❝✐❛ ❞❡ G é ❛ ♠❛tr✐③

Adj(G) = (aij)n×n✱ t❛❧ q✉❡

aij =

1, s❡ i6=j e{i, j} ∈E

0, s❡ nao

▲❡♠❛ ✶✳✶ ❆ ♠❛tr✐③ ❞❡ ❛❞❥❛❝ê♥❝✐❛ ❞❡ ✉♠ ❣r❛❢♦ é s✐♠étr✐❝❛✱ ❝♦♠ ③❡r♦s ♥❛ ❞✐❛❣♦♥❛❧ ♣r✐♥❝✐♣❛❧✳

Pr♦✈❛✳

◆❛s ♥♦t❛çõ❡s ❞❛ ❞❡✜♥✐çã♦ ❛❝✐♠❛✱ t❡♠♦s ♣❛r❛i6=j ❡♠ In q✉❡

a_ij = 1⇔ {i, j} ∈E ⇔ {j, i} ∈E ⇔a_ji = 1;

♣♦rt❛♥t♦✱ Adj(G) é s✐♠étr✐❝❛✳ ❖ r❡st♦ é ✐♠❡❞✐❛t♦ ❛ ♣❛rt✐r ❞❛ ❞❡✜♥✐çã♦ ❛❝✐♠❛✳

❊①❡♠♣❧♦ ✶✳✷ ❚♦♠❡♠♦s ❝♦♠♦ ❡①❡♠♣❧♦ ♦ ❣r❛❢♦ ❞❛ ✜❣✉r❛ ✶✳✶ ❛❝✐♠❛✱ ❛ ♠❛tr✐③ ❞❡ ❛❞❥❛❝ê♥❝✐❛ ❞❡st❡ ❣r❛❢♦ é✿

M =



  

0 1 1 1 1 0 1 0 1 1 0 0 1 0 0 0



  

❉❡✜♥✐çã♦ ✶✳✽ ❯♠ ❣r❛❢♦ H é ✉♠ s✉❜❣r❛❢♦ ❞❡ ✉♠ ❣r❛❢♦ G s❡ V(H) ⊂ V(G) ❡

E(H)⊂E(G)✳ ❖ ❣r❛❢♦ H é ✉♠ s✉❜❣r❛❢♦ ❣❡r❛❞♦r ❞❡ G s❡ H ❢♦r ✉♠ s✉❜❣r❛❢♦ ❞❡ G t❛❧ q✉❡ V(H) =V(G)✳

●r❛❢♦s ❈❛♣ít✉❧♦ ✶

❋✐❣✉r❛ ✶✳✷✿ ❖ ❣r❛❢♦ ● ❡ s❡✉ ❝♦♠♣❧❡♠❡♥t❛r✳

❉❡✜♥✐çã♦ ✶✳✾ ❙❡ G = (V;E) é ✉♠ ❣r❛❢♦✱ s❡✉ ❣r❛❢♦ ❝♦♠♣❧❡♠❡♥t❛r é ♦ ❣r❛❢♦ ¯

G= (V;Ec_{)✱ ♦♥❞❡ E}c _{❞❡♥♦t❛ ♦ ❝♦♠♣❧❡♠❡♥t❛r ❞❡ ❊✳}

❉❡✜♥✐çã♦ ✶✳✶✵ ❯♠ ❣r❛❢♦ G= (V;E) é ❜✐♣❛rt✐❞♦ s❡ ♣✉❞❡r♠♦s ❡s❝r❡✈❡r

V =V₁∪V₂✱ ♦♥❞❡V₁ ❡ V₂ sã♦ ❝♦♥❥✉♥t♦s ❞✐s❥✉♥t♦s✱ ♥ã♦ ✈❛③✐♦s ❡ ✐♥❞❡♣❡♥❞❡♥t❡s✳

◆❛ t❡♦r✐❛ ❞♦s ❣r❛❢♦s✱ ✉♠ ❝♦♥❥✉♥t♦ ✐♥❞❡♣❡♥❞❡♥t❡ ❞❡ ✉♠ ❣r❛❢♦Gé ✉♠ ❝♦♥❥✉♥t♦ S ❞❡ ✈ért✐❝❡s ❞❡ G t❛❧ q✉❡ ♥ã♦ ❡①✐st❡♠ ❞♦✐s ✈ért✐❝❡s ❛❞❥❛❝❡♥t❡s ❝♦♥t✐❞♦s ❡♠ S✳ ❊♠

♦✉tr❛s ♣❛❧❛✈r❛s✱ s❡ a ❡ b sã♦ ✈ért✐❝❡s q✉❛✐sq✉❡r ❞❡ ✉♠ ❝♦♥❥✉♥t♦ ✐♥❞❡♣❡♥❞❡♥t❡✱ ♥ã♦

❤á ❛r❡st❛ ❡♥tr❡ a ❡ b✳

❋✐❣✉r❛ ✶✳✸✿ ●r❛❢♦ ❜✐♣❛rt✐❞♦✳

❉❡✜♥✐çã♦ ✶✳✶✶ ❯♠ ❣r❛❢♦ é r✲r❡❣✉❧❛r s❡ t♦❞♦s ♦s ✈ért✐❝❡s ❞❡ G t❡♠ ❣r❛✉ r✳

❋✐❣✉r❛ ✶✳✹✿ ●r❛❢♦ ✸✲r❡❣✉❧❛r✳

●r❛❢♦s ❈❛♣ít✉❧♦ ✶

✶✳✷ ❆❧❣✉♥s ❊①❡♠♣❧♦s ❞❡ ●r❛❢♦s

❊st❛ s❡çã♦ ❢♦✐ ❡❧❛❜♦r❛❞❛ ❛ ♣❛rt✐r ❞❛ s❡❣✉✐♥t❡ r❡❢❡rê♥❝✐❛ ❜✐❜❧✐♦❣rá✜❝❛✿ ❬✹❪✳

❊①❡♠♣❧♦ ✶✳✸ ❖s ✈ért✐❝❡s ❞♦ ❣r❛❢♦ sã♦ ❛s ❝❛s❛s ❞❡ ✉♠ t❛❜✉❧❡✐r♦ ❞❡ ①❛❞r❡③ ✭❣❡♥❡✲ r❛❧✐③❛❞♦✮ ❝♦♠ t ❧✐♥❤❛s ❡ t ❝♦❧✉♥❛s ✭♥♦ t❛❜✉❧❡✐r♦ ✉s✉❛❧✱ t ✈❛❧❡ ✽✮✳ ❉♦✐s ✈ért✐❝❡s sã♦ ❛❞❥❛❝❡♥t❡s s❡ ✉♠❛ r❛✐♥❤❛ ❞♦ ❥♦❣♦ ❞❡ ①❛❞r❡③ ♣♦❞❡ s❛❧t❛r ❞❡ ✉♠ ❞❡❧❡s ♣❛r❛ ♦ ♦✉tr♦ ❡♠ ✉♠ só ♠♦✈✐♠❡♥t♦✳ ❊ss❡ é ♦ ❣r❛❢♦ ❞♦s ♠♦✈✐♠❡♥t♦s ❞❛ r❛✐♥❤❛✱ ♦✉ s✐♠♣❧❡s♠❡♥t❡ ♦ ❣r❛❢♦ ❞❛ r❛✐♥❤❛✳ P❛r❛ ❞❡✐①❛r ❝❧❛r❛s ❛s ❞✐♠❡♥sõ❡s ❞♦ t❛❜✉❧❡✐r♦✱ ♣♦❞❡♠♦s ❞✐③❡r q✉❡ ❡ss❡ é ♦ ❣r❛❢♦ ❞❛ r❛✐♥❤❛ t ♣♦r t✳ P♦r ❛♥❛❧♦❣✐❛✱ ❞❡✜♥❡♠✲s❡ ♦ ❣r❛❢♦ ❞♦ r❡✐✱ ♦ ❣r❛❢♦ ❞♦ ❜✐s♣♦✱ ♦ ❣r❛❢♦ ❞♦ ❝❛✈❛❧♦ ❡ ♦ ❣r❛❢♦ ❞❛ t♦rr❡ t ♣♦r t✳

❋✐❣✉r❛ ✶✳✺✿ ❚❛❜✉❧❡✐r♦s ❞❡ ①❛❞r❡③ ✽ ♣♦r ✽✳ ❆ ✜❣✉r❛ ❡sq✉❡r❞❛ r❡♣r❡s❡♥t❛ ♦ ❣r❛❢♦ ❞❛ r❛✐♥❤❛ ❡ ❛ ❞❛ ❞✐r❡✐t❛ r❡♣r❡s❡♥t❛ ♦ ❣r❛❢♦ ❞♦ ❝❛✈❛❧♦✳

❊①❡♠♣❧♦ ✶✳✹ ❖ ❣r❛❢♦ ❞♦s ❡st❛❞♦s ❞♦ ❇r❛s✐❧ é ❞❡✜♥✐❞♦ ❛ss✐♠✿ ❝❛❞❛ ✈ért✐❝❡ é ✉♠ ❞♦s ❡st❛❞♦s ❞❛ ❘❡♣ú❜❧✐❝❛ ❋❡❞❡r❛t✐✈❛ ❞♦ ❇r❛s✐❧❀ ❞♦✐s ❡st❛❞♦s sã♦ ❛❞❥❛❝❡♥t❡s s❡ tê♠ ✉♠❛ ❢r♦♥t❡✐r❛ ❝♦♠✉♠✳

❋✐❣✉r❛ ✶✳✻✿ ●r❛❢♦ ❞♦s ❡st❛❞♦s ❞♦ ❇r❛s✐❧✳

●r❛❢♦s ❈❛♣ít✉❧♦ ✶

❊①❡♠♣❧♦ ✶✳✺ ❖s ❤✐❞r♦❝❛r❜♦♥❡t♦s ❝♦♥❤❡❝✐❞♦s ❝♦♠♦ ❛❧❝❛♥♦s tê♠ ❢ór♠✉❧❛ q✉í♠✐❝❛

CpH2p+2 ✱ ♦♥❞❡ C ❡ H r❡♣r❡s❡♥t❛♠ ♠♦❧é❝✉❧❛s ❞❡ ❝❛r❜♦♥♦ ❡ ❤✐❞r♦❣ê♥✐♦ r❡s♣❡❝t✐✈❛✲

♠❡♥t❡✳ ❆s ♠♦❧é❝✉❧❛s ❞❡ ❛❧❝❛♥♦s ♣♦❞❡♠ s❡r r❡♣r❡s❡♥t❛❞❛s ♣♦r ❣r❛❢♦s ❝♦♠♦ ♦s ❞❛ ✜❣✉r❛ ❛❜❛✐①♦✳

❋✐❣✉r❛ ✶✳✼✿ ❊t❛♥♦ ✭C2H6✮✱ ❜✉t❛♥♦ ✭C4H10✮ ❡ ✐s♦❜✉t❛♥♦ ✭C4H10✮✳ ❖s ✈ért✐❝❡s ❡♠

q✉❡ ✐♥❝✐❞❡ ✉♠❛ só ❛r❡st❛ r❡♣r❡s❡♥t❛♠ át♦♠♦s ❞❡ ❤✐❞r♦❣ê♥✐♦✱ ♦s ❞❡♠❛✐s r❡♣r❡s❡♥t❛♠ át♦♠♦s ❞❡ ❝❛r❜♦♥♦✳

❈❛♣ít✉❧♦ ✷

P❛ss❡✐♦s✱ ❈❛♠✐♥❤♦s✱ ❈✐❝❧♦s ❡ ➪r✈♦r❡s

✷✳✶ ■❞❡✐❛s ❇ás✐❝❛s

❊st❛ s❡çã♦ ❢♦✐ ❡❧❛❜♦r❛❞❛ ❛ ♣❛rt✐r ❞❛s s❡❣✉✐♥t❡s r❡❢❡rê♥❝✐❛s ❜✐❜❧✐♦❣rá✜✲ ❝❛s✿ ❬✶❪✱ ❬✷❪ ❡ ❬✸❪✳

❉❡✜♥✐çã♦ ✷✳✶ ❯♠ ♣❛ss❡✐♦ ❞❡ ❝♦♠♣r✐♠❡♥t♦k ≥1❡♠ ✉♠ ❣r❛❢♦Gé ✉♠❛ s❡q✉ê♥✲

❝✐❛ P = (u₀, u₁, ..., uk)❞❡ ✈ért✐❝❡s ✭♥ã♦ ♥❡❝❡ss❛r✐❛♠❡♥t❡ ❞✐st✐♥t♦s✮ ❞❡ G t❛❧ q✉❡ui−1

é ❛❞❥❛❝❡♥t❡ ❛ ui ♣❛r❛1≤i≤k✳ ❖ ♣❛ss❡✐♦ P ❛❝✐♠❛ é ❢❡❝❤❛❞♦ s❡u0 =uk✱ ♣♦❞❡♠♦s

❝❤❛♠❛r ♦ ♣❛ss❡✐♦ ❢❡❝❤❛❞♦ ❞❡ ❝✐r❝✉✐t♦✳

❋✐❣✉r❛ ✷✳✶✿ ❯♠ ♣❛ss❡✐♦ ❡ ✉♠ ❝✐r❝✉✐t♦✳

❉❡✜♥✐çã♦ ✷✳✷ ❯♠ ❣r❛❢♦ é ❝♦♥❡①♦ s❡ ❡①✐st✐r ✉♠ ♣❛ss❡✐♦ ❡♥tr❡ ❞♦✐s q✉❛✐sq✉❡r ❞❡ s❡✉s ✈ért✐❝❡s✳ ❯♠ ❣r❛❢♦ q✉❡ ♥ã♦ é ❝♦♥❡①♦ é ❞✐t♦ ❞❡s❝♦♥❡①♦✳

❆ ♣r♦♣♦s✐ç❛♦ s❡❣✉✐♥t❡ ❞❡✜♥❡ ✉♠ ❝r✐tér✐♦ s✐♠♣❧❡s ♣❛r❛ ✉♠ ❣r❛❢♦ s❡r ❝♦♥❡①♦✳ P❛r❛ ♦ ❡♥✉♥❝✐❛❞♦ ❞❡❧❛✱ ♣r❡❝✐s❛♠♦s ✐♥tr♦❞✉③✐r ❛ s❡❣✉✐♥t❡ ♥♦t❛çã♦✿ ❞❛❞♦ ✉♠ ❣r❛❢♦ G = (V;E)✱ ♦ ❣r❛✉ ♠í♥✐♠♦ ❞❡ Gé ♦ ✐♥t❡✐r♦ ♥ã♦ ♥❡❣❛t✐✈♦

δ(G) =min{dG(u);u∈V(G)}. (2.1)

Pr♦♣♦s✐çã♦ ✷✳✶ ❙❡ G é ✉♠ ❣r❛❢♦ ❝♦♠ n ✈ért✐❝❡s ❡ t❛❧ q✉❡ δ(G)≥ ⌊n

2⌋✱ ❡♥tã♦ G é

❝♦♥❡①♦✳

P❛ss❡✐♦s✱ ❈❛♠✐♥❤♦s✱ ❈✐❝❧♦s ❡ ➪r✈♦r❡s ❈❛♣ít✉❧♦ ✷

Pr♦✈❛✳

❙❡n = 1 ♦✉ n = 2✱ ♥ã♦ ❤á ♥❛❞❛ ❛ ❢❛③❡r✳ ❙✉♣♦♥❤❛ n >2 ❡ s❡❥❛♠ u ❡ v ✈ért✐❝❡s

❞✐st✐♥t♦s ❞❡ G✳ ❈❛s♦ u ❡ v s❡❥❛♠ ❛❞❥❛❝❡♥t❡s✱ ♥❛❞❛ ❤á ❛ ❢❛③❡r✳ ❉♦ ❝♦♥trár✐♦✱ ❝♦♠♦

dG(u) +dG(v)≥2⌊₂n⌋> n−2 = #(V(G) \ {u, v})✱

♦ ♣r✐♥❝í♣✐♦ ❞❛ ❝❛s❛ ❞♦s ♣♦♠❜♦s ✶ _{❣❛r❛♥t❡ ❛ ❡①✐stê♥❝✐❛ ❞❡ ✉♠ ✈ért✐❝❡ w 6= u, v✱}

❛❞❥❛❝❡♥t❡ ❛ ❛♠❜♦s u ❡v✳ ▲♦❣♦✱ ♦ ♣❛ss❡✐♦ (u, w, v) ❧✐❣❛ u ❛v✳

❉❡✜♥✐çã♦ ✷✳✸ ❯♠ ❝❛♠✐♥❤♦ ❡♠ ✉♠ ❣r❛❢♦ é ✉♠ ♣❛ss❡✐♦ t❛❧ q✉❡ t♦❞♦s ♦s s❡✉s ✈ér✲ t✐❝❡s sã♦ ❞✐st✐♥t♦s✳

❆♣❡s❛r ❞❡ t♦❞♦ ❝❛♠✐♥❤♦ s❡r ✉♠ ♣❛ss❡✐♦✱ é ✐♠❡❞✐❛t♦ q✉❡✱ ❡♠ ❣❡r❛❧✱ ❤á ♣❛ss❡✐♦s q✉❡ ♥ã♦ sã♦ ❝❛♠✐♥❤♦s✳ ◆♦ ❡♥t❛♥t♦✱ ♥ã♦ é ❞✐❢í❝✐❧ ♥♦s ❝♦♥✈❡♥❝❡r♠♦s q✉❡ s❡ ❤á ✉♠ ♣❛ss❡✐♦ ❡♥tr❡ ❞♦✐s ✈ért✐❝❡s ❞❡ ✉♠ ❣r❛❢♦✱ ❡♥tã♦ t❛♠❜é♠ ❤á ✉♠ ❝❛♠✐♥❤♦ ❡♥tr❡ ❡❧❡s✳ ■st♦ ♠♦t✐✈❛ ♦ t❡♦r❡♠❛ ❛❜❛✐①♦✳

❚❡♦r❡♠❛ ✷✳✶ ❙❡ ❤♦✉✈❡r ✉♠ ♣❛ss❡✐♦ ❞♦ ✈ért✐❝❡ ② ❛♦ ✈ért✐❝❡ ③ ♥♦ ❣r❛❢♦ ●✱ ♦♥❞❡ ② ♥ã♦ é ✐❣✉❛❧ ❛ ③✱ ❡♥tã♦ ❤á ✉♠ ❝❛♠✐♥❤♦ ❡♠ ● ❝♦♠ ♣r✐♠❡✐r♦ ✈ért✐❝❡ ② ❡ ú❧t✐♠♦ ✈ért✐❝❡ ③✳ ✭❊♠ ♦✉tr❛s ♣❛❧❛✈r❛s✱ s❡ ❤á ✉♠ ♣❛ss❡✐♦ ❡♥tr❡ ❞♦✐s ✈ért✐❝❡s ❞❡ ✉♠ ❣r❛❢♦✱ ❤á t❛♠❜é♠ ✉♠ ❝❛♠✐♥❤♦ ❡♥tr❡ ♦s ♠❡s♠♦s✳✮

Pr♦✈❛✳

❈❤❛♠❡♠♦s ♦ ♣❛ss❡✐♦✲②③ ❞❡

W1 =x0, a1, x1, a2, ..., an, xn,

❝♦♠♦ ❛❝✐♠❛✱ ♦♥❞❡ y = x0 ❡ z = xn✳ ❙❡ ♦s ✈ért✐❝❡s x0, x1, ..., xn sã♦ t♦❞♦s

❞✐❢❡r❡♥t❡s✱ W₁ é ✉♠ ❝❛♠✐♥❤♦ ❡ ♥ã♦ ❤á ♥❛❞❛ ❛ ♣r♦✈❛r✳ ❙❡ ♥ã♦✱ s❡❧❡❝✐♦♥❡ ✉♠ ✈ért✐❝❡

q✉❡ ❛♣❛r❡❝❡ ❞✉❛s ✈❡③❡s✿ ❝❤❛♠❡ xi =xj✱ ♦♥❞❡i < j✳ ❊s❝r❡✈❛

W2 =x0, a1, x1, ..., xi, aj+1, xj+1, ..., an, xn.

❊♥tã♦W2 é ✉♠ ♣❛ss❡✐♦ ❞❡ ② ♣❛r❛ ③ ❡ é ♠❛✐s ❝✉rt♦ ❞♦ q✉❡ W1✳

❙❡ W2 ♥ã♦ ❝♦♥té♠ ✈ért✐❝❡ r❡♣❡t✐❞♦✱ ❡♥tã♦ é ♥❡❝❡ss❛r✐❛♠❡♥t❡ ❝❛♠✐♥❤♦✳ ❈❛s♦

❝♦♥trár✐♦✱ s❡❧❡❝✐♦♥❡ ✉♠ ✈ért✐❝❡ r❡♣❡t✐❞♦✱ ❡ ♣r♦❝❡❞❛ ❝♦♠♦ ❛❝✐♠❛✳ ▼❛✐s ✉♠❛ ✈❡③ ✉♠ ♣❛ss❡✐♦ ♠❛✐s ❝✉rt♦ é ❝♦♥str✉í❞♦✳

✶_{❖ ♣r✐♥❝í♣✐♦ ❞❛ ❝❛s❛ ❞♦s ♣♦♠❜♦s✱ t❛♠❜é♠ ❝♦♥❤❡❝✐❞♦ ❝♦♠♦ ♣r✐♥❝í♣✐♦ ❞❛s ❣❛✈❡t❛s ♦✉ ♣r✐♥❝í♣✐♦ ❞❡}

❉✐r✐❝❤❧❡t✱ ♣♦❞❡ s❡r ❡♥✉♥❝✐❛❞♦ ❞❛ s❡❣✉✐♥t❡ ❢♦r♠❛✿ ❙❡ t❡♠♦s ❞❡ ❞✐str✐❜✉✐r ♥ ♣♦♠❜♦s ❡♠ ♥✲✶ ❣❛✐♦❧❛s✱ ❡♥tã♦ ❛♦ ♠❡♥♦s ✉♠❛ ❞❛s ❣❛✐♦❧❛s ❝♦♥t❡rá✱ ♥♦ ♠í♥✐♠♦✱ ❞♦✐s ♣♦♠❜♦s✳

P❛ss❡✐♦s✱ ❈❛♠✐♥❤♦s✱ ❈✐❝❧♦s ❡ ➪r✈♦r❡s ❈❛♣ít✉❧♦ ✷

❊st❡ ♣r♦❝❡ss♦ ❞❡✈❡ ♣❛r❛r ♥✉♠ ❞❛❞♦ ♠♦♠❡♥t♦✱ ✉♠❛ ✈❡③ q✉❡ ❝❛❞❛ ♣❛ss❡✐♦ é ♠❡♥♦r ❞♦ q✉❡ ♦ ❛♥t❡r✐♦r ❡ ♦ ❝♦♠♣r✐♠❡♥t♦ ♥ã♦ ♣♦❞❡ s❡r ✐♥❢❡r✐♦r ❛ ✶✳ ❆ss✐♠✱ ♣❛r❛ ❛❧❣✉♥s

k✱ Wk ♥ã♦ ♣♦❞❡ s❡r r❡❞✉③✐❞❛ ❡♠ ❝♦♠♣r✐♠❡♥t♦✳ P♦rt❛♥t♦ Wk ♥ã♦ ❝♦♥té♠ ✈ért✐❝❡

r❡♣❡t✐❞♦ ❡ é ♥❡❝❡ss❛r✐❛♠❡♥t❡ ❝❛♠✐♥❤♦✳

❉❡✜♥✐çã♦ ✷✳✹ ❯♠ ❦✲❝✐❝❧♦ ❡♠ ✉♠ ❣r❛❢♦ G é ✉♠ ♣❛ss❡✐♦ ❢❡❝❤❛❞♦

(u0, u1, ..., uk−1, u0)

❡♠ G✱ ❞❡ ❝♦♠♣r✐♠❡♥t♦ k≥3 ❡ t❛❧ q✉❡ (u₀, u₁, ..., u_k−1) é ✉♠ ❝❛♠✐♥❤♦✳

❉❡✜♥✐çã♦ ✷✳✺ ❯♠ ❣r❛❢♦ G é ❜✐♣❛rt✐❞♦ s❡ t♦❞♦ ❝✐❝❧♦ ❡♠ G t❡♠ ❝♦♠♣r✐♠❡♥t♦ ♣❛r✳

Pr♦✈❛✳

❙✉♣♦♥❤❛ ✐♥✐❝✐❛❧♠❡♥t❡ q✉❡Gé ❜✐♣❛rt✐❞♦✱ ❝♦♠ ❝♦♥❥✉♥t♦s ✐♥❞❡♣❡♥❞❡♥t❡s ❞❡ ✈ért✐❝❡s V₁ ❡ V₂✳ ❙❡ C = (u₀, u₁, u₂, ..., uk, u0) é ✉♠ ❝✐❝❧♦ ❡♠ G✱ ❡♥tã♦ ❛ ✐♥❞❡♣❡♥❞ê♥❝✐❛ ❞♦s

❝♦♥❥✉♥t♦s V1 ❡ V2 ❣❛r❛♥t❡ q✉❡ ♦s ✈ért✐❝❡s ui ❡ ui+1 ❛❧t❡r♥❛♠✲s❡ ❡♥tr❡ ♦s ❝♦♥❥✉♥t♦s

V₁ ❡ V₂✳ ❊♠ ♣❛rt✐❝✉❧❛r✱ ♦ ❝❛♠✐♥❤♦ (u₀, u₁, ..., uk) ❞á k ❛❧t❡r♥â♥❝✐❛s ❡♥tr❡ V1 ❡ V2

♣❛r❛✱ ♣❛rt✐♥❞♦ ❞❡u₀✱ ❝❤❡❣❛r♠♦s ❛uk✳ ▼❛s ❝♦♠♦uk❡ u0 sã♦ ❛❞❥❛❝❡♥t❡s✱ ♥ã♦ ♣♦❞❡♠

♣❡rt❡♥❝❡r ❛ ✉♠ ♠❡s♠♦ ❞♦s ❝♦♥❥✉♥t♦s V1✱ V2 ❡✱ ❞❛í✱ k ❞❡✈❡ s❡r í♠♣❛r✳ P♦rt❛♥t♦✱ ♦

❝♦♠♣r✐♠❡♥t♦ ❞❡ k+ 1 ❞❡ C é ♣❛r✳ ❘❡❝✐♣r♦❝❛♠❡♥t❡✱ s✉♣♦♥❤❛ q✉❡ t♦❞♦ ❝✐❝❧♦ ❡♠ G

t❡♥❤❛ ❝♦♠♣r✐♠❡♥t♦ ♣❛r ❡ ♠♦str❡♠♦s q✉❡ G é ❜✐♣❛rt✐❞♦ ❢❛③❡♥❞♦ ✐♥❞✉çã♦ s♦❜r❡ s❡✉

♥ú♠❡r♦ ❞❡ ❛r❡st❛s✳ ❙❡ G ❝♦♥t✐✈❡r ❛❧❣✉♠ ❝✐❝❧♦ C✱ r❡t✐r❡ ✉♠❛ ❛r❡st❛ ǫ={u, v} ❞❡ C✳

❈♦♠♦ ♥ã♦ ❢♦r❛♠ ❝r✐❛❞♦s ♥♦✈♦s ❝✐❝❧♦s✱ G−ǫ❛✐♥❞❛ é t❛❧ q✉❡ t♦❞♦s ♦s s❡✉s ❝✐❝❧♦s tê♠

❝♦♠♣r✐♠❡♥t♦ ♣❛r✳ P♦rt❛♥t♦✱ ♣❡❧❛ ❤✐♣ót❡s❡ ❞❡ ✐♥❞✉çã♦✱ G−ǫ é ❜✐♣❛rt✐❞♦✱ ❞✐❣❛♠♦s

❝♦♠ ❝♦♥❥✉♥t♦s ✐♥❞❡♣❡♥❞❡♥t❡s ❞❡ ✈ért✐❝❡s V1 ❡ V2✳ ◆♦t❡ ❛❣♦r❛ q✉❡ ♦ ❝❛♠✐♥❤♦ C −ǫ

❡♠ G−ǫ❧✐❣❛u❛v ❡ t❡♠ ❝♦♠♣r✐♠❡♥t♦ í♠♣❛r❀ ✉♠❛ ✈❡③ q✉❡ s❡✉s ✈ért✐❝❡s s❡ ❛❧t❡r♥❛♠

❡♥tr❡V₁ ❡V₂✱ s❡❣✉❡ q✉❡ u∈V₁ ❡v ∈V₂✱ ♦✉ ✈✐❝❡✲✈❡rs❛✳ ▲♦❣♦✱Gt❛♠❜é♠ é ❜✐♣❛rt✐❞♦✱

t❛♠❜é♠ ❝♦♠ ❝♦♥❥✉♥t♦s ✐♥❞❡♣❡♥❞❡♥t❡s ❞❡ ✈ért✐❝❡s V1 ❡ V2✳

P❛ss❡✐♦s✱ ❈❛♠✐♥❤♦s✱ ❈✐❝❧♦s ❡ ➪r✈♦r❡s ❈❛♣ít✉❧♦ ✷

✷✳✷ P❛ss❡✐♦s ❊✉❧❡r✐❛♥♦s

❊st❛ s❡çã♦ ❢♦✐ ❡❧❛❜♦r❛❞❛ ❛ ♣❛rt✐r ❞❛s s❡❣✉✐♥t❡s r❡❢❡rê♥❝✐❛s ❜✐❜❧✐♦❣rá✜✲ ❝❛s✿ ❬✶❪✱ ❬✷❪ ❡ ❬✸❪✳

❉❡✜♥✐çã♦ ✷✳✻ ❯♠ ♣❛ss❡✐♦ ❊✉❧❡r✐❛♥♦ ❡♠ ✉♠ ❣r❛❢♦ ❝♦♥❡①♦ é ✉♠ ♣❛ss❡✐♦ ❢❡❝❤❛❞♦ q✉❡ ✉s❛ ❝❛❞❛ ❛r❡st❛ ❞♦ ❣r❛❢♦ ❡①❛t❛♠❡♥t❡ ✉♠❛ ✈❡③✳ ❯♠ ❣r❛❢♦ ❝♦♥❡①♦ é ❊✉❧❡r✐❛♥♦ s❡ ❝♦♥t✐✈❡r ✉♠ ♣❛ss❡✐♦ ❊✉❧❡r✐❛♥♦✳

❙❡❥❛ G ✉♠ ❣r❛❢♦ ❊✉❧❡r✐❛♥♦ ❡P ✉♠ ♣❛ss❡✐♦ ❊✉❧❡r✐❛♥♦ ❡♠G✳ ❋✐①❛❞♦ ❛r❜✐tr❛r✐❛✲

♠❡♥t❡ ✉♠ ✈ért✐❝❡ u ❞❡ G✱ s❡ k ≥ 1 é ♦ ♥ú♠❡r♦ ❞❡ ♦❝♦rrê♥❝✐❛s ❞❡ u ❡♠ P✱ ❡♥tã♦ é

❝❧❛r♦ q✉❡ dG(u) = 2k✱ ✉♠ ♥ú♠❡r♦ ♣❛r✳ P♦rt❛♥t♦✱ ✉♠❛ ❝♦♥❞✐çã♦ ♥❡❝❡ssár✐❛ ♣❛r❛ q✉❡

✉♠ ❣r❛❢♦ ❝♦♥❡①♦ G s❡❥❛ ❊✉❧❡r✐❛♥♦ é q✉❡ t♦❞♦s ♦s ✈ért✐❝❡s ❞❡ G t❡♥❤❛♠ ❣r❛✉ ♣❛r✳

❖ t❡♦r❡♠❛ s❡❣✉✐♥t❡ ❣❛r❛♥t❡ q✉❡ t❛❧ ❝♦♥❞✐çã♦ t❛♠❜é♠ é s✉✜❝✐❡♥t❡ ♣❛r❛ ✉♠ ❣r❛❢♦ s❡r ❊✉❧❡r✐❛♥♦✳

▲❡♠❛ ✷✳✶ ❙❡ G é ✉♠ ❣r❛❢♦ ❝♦♥❡①♦ ♥♦ q✉❛❧ t♦❞♦ ✈ért✐❝❡ t❡♠ ❣r❛✉ ♣❛r✱ ❡♥tã♦ G

❝♦♥té♠ ✉♠ ♣❛ss❡✐♦ ❢❡❝❤❛❞♦ q✉❡ ♥ã♦ ❛tr❛✈❡ss❛ ✉♠❛ ♠❡s♠❛ ❛r❡st❛ ❞✉❛s ✈❡③❡s✳

Pr♦✈❛✳

❊s❝♦❧❤❛ ✉♠ ✈ért✐❝❡ u0 ❞❡ G ❡✱ ❛ ♣❛rt✐r ❞❡ u0✱ ❝r✐❡ ✉♠ ♣❛ss❡✐♦ ❡♠ G r❡♣❡t✐♥❞♦✱

❡♥q✉❛♥t♦ ♣♦ssí✈❡❧✱ ♦ s❡❣✉✐♥t❡ ❛❧❣♦r✐t♠♦✿ ❡st❛♥❞♦ ❡♠ ✉♠ ✈ért✐❝❡ uk−1 ❞❡ G✱ k ≥1✱

❡s❝♦❧❤❛ ✉♠ ✈ért✐❝❡uk ❛❞❥❛❝❡♥t❡ ❛uk−1 ❡ t❛❧ q✉❡ ❛ ❛r❡st❛{uk−1, uk}❛✐♥❞❛ ♥ã♦ t❡♥❤❛

s✐❞♦ ❛tr❛✈❡ss❛❞❛✳ P❛r❛ ♠♦str❛r q✉❡ ❡ss❡ ❛❧❣♦r✐t♠♦ ❣❡r❛ ✉♠ ♣❛ss❡✐♦ ❢❡❝❤❛❞♦ ❡♠Gq✉❡

♥ã♦ ❛tr❛✈❡ss❛ ✉♠❛ ♠❡s♠❛ ❛r❡st❛ ❞✉❛s ✈❡③❡s✱ s✉♣♦♥❤❛ q✉❡✱ ❛♣ós ❛ ❡①❡❝✉çã♦ ❞❡ ✉♠ ❝❡rt♦ ♥ú♠❡r♦ ❞❡ ♣❛ss♦s ❡st❡❥❛♠♦s ❡♠ ✉♠ ✈ért✐❝❡ uk ❞❡G✳ ❍á ❞✉❛s ♣♦ss✐❜✐❧✐❞❛❞❡s✿

✶✳ uk =u0✿ ♥❛❞❛ ❤á ❛ ❢❛③❡r✱ ♣♦✐s ♦ ♣❛ss❡✐♦ ❞❡s❡❥❛❞♦ é(u0, u1, ..., uk)✳

✷✳ uk 6=u0✿ s✉♣♦♥❤❛ q✉❡ ♦ ✈ért✐❝❡ uk ❥á ❛♣❛r❡❝❡✉ ✭❝♦♠ í♥❞✐❝❡s ♠❡♥♦r❡s✮ ♦✉tr❛s

l ✈❡③❡s❀ ❡♥tã♦ ✉s❛♠♦s ❡①❛t❛♠❡♥t❡ 2l + 1 ❞❡♥tr❡ ❛s ❛r❡st❛s ✐♥❝✐❞❡♥t❡s ❡♠ uk

✭❞✉❛s ❛r❡st❛s ♣❛r❛ ❝❛❞❛ ✉♠❛ ❞❛s l ♦❝♦rrê♥❝✐❛s ❛♥t❡r✐♦r❡s✱ ♠❛✐s ✉♠❛ ❛r❡st❛

♣❛r❛ ❝❤❡❣❛r ❡♠ uk ❛ ♣❛rt✐r ❞❡ uk−1✮✳ ▼❛s ❝♦♠♦ uk t❡♠ ❣r❛✉ ♣❛r✱ ❡①✐st❡ ♣❡❧♦

♠❡♥♦s ✉♠❛ ❛r❡st❛ ✐♥❝✐❞❡♥t❡ ❡♠ t❛❧ ✈ért✐❝❡ ❡ ❛✐♥❞❛ ♥ã♦ ✉t✐❧✐③❛❞❛❀ ♣♦rt❛♥t♦✱ ♦ ❛❧❣♦r✐t♠♦ ♥ã♦ ♣❛r❛ ❡♠ uk✳

P♦r ✜♠✱ ❝♦♠♦ ♦ ♥ú♠❡r♦ ❞❡ ❛r❡st❛s ❞❡Gé ✜♥✐t♦✱ ❛ ❞✐s❝✉ssã♦ ❛❝✐♠❛ ❣❛r❛♥t❡ q✉❡ ♦

❛❧❣♦r✐t♠♦ ♣❛r❛ ❡♠ u0 ❡✱ ❛♦ ♣❛r❛r✱ t❡r❡♠♦s ✉♠ ♣❛ss❡✐♦ ❢❡❝❤❛❞♦ Gq✉❡ ♥ã♦ ❛tr❛✈❡ss❛

✉♠❛ ♠❡s♠❛ ❛r❡st❛ ❞✉❛s ✈❡③❡s✳

❚❡♦r❡♠❛ ✷✳✷ ✭❊✉❧❡r✮ ❯♠ ❣r❛❢♦ ❝♦♥❡①♦ ❡ ♥ã♦ tr✐✈✐❛❧Gé ❊✉❧❡r✐❛♥♦ s❡ ❡ só s❡ t♦❞♦s

♦s s❡✉s ✈ért✐❝❡s tê♠ ❣r❛✉ ♣❛r✳

P❛ss❡✐♦s✱ ❈❛♠✐♥❤♦s✱ ❈✐❝❧♦s ❡ ➪r✈♦r❡s ❈❛♣ít✉❧♦ ✷

Pr♦✈❛✳

❏á ✈✐♠♦s q✉❡✱ s❡ Gé ❊✉❧❡r✐❛♥♦✱ ❡♥tã♦ t♦❞♦s ♦s s❡✉s ✈ért✐❝❡s tê♠ ❣r❛✉ ♣❛r✳ P❛r❛

❡st❛❜❡❧❡❝❡r ❛ r❡❝í♣r♦❝❛✱ ❢❛ç❛♠♦s ✐♥❞✉çã♦ ❡♠ r❡❧❛çã♦ ❛♦ ♥ú♠❡r♦ ❞❡ ❛r❡st❛s ❞❡ G✳ ❆

❝♦♥❞✐çã♦ ❞♦ ❡♥✉♥❝✐❛❞♦ ✐♠♣❧✐❝❛ q✉❡Gt❡♠ ♣❡❧♦ ♠❡♥♦s três ❛r❡st❛s❀ ❛❞❡♠❛✐s✱ s❡Gt✐✲

✈❡r ❡①❛t❛♠❡♥t❡ três ❛r❡st❛s✱ ❡♥tã♦Gé ✐s♦♠♦r❢♦ ❛K3✱ ♦ q✉❛❧ é ❝❧❛r❛♠❡♥t❡ ❊✉❧❡r✐❛♥♦✳

❙✉♣♦♥❤❛ ❛❣♦r❛ q✉❡ G t❡♠ n >3 ❛r❡st❛s✱ ❡ q✉❡ ♦ t❡♦r❡♠❛ é ✈❡r❞❛❞❡✐r♦ ♣❛r❛ ❣r❛❢♦s

❝♦♥❡①♦s ❝♦♠ ♠❡♥♦s ❞❡ n ❛r❡st❛s✱ ♥♦s q✉❛✐s t♦❞♦s ♦s ✈ért✐❝❡s tê♠ ❣r❛✉ ♣❛r✳ P❡❧♦

❧❡♠❛ ❛♥t❡r✐♦r✱ ♣♦❞❡♠♦s t♦♠❛r ❡♠ G ✉♠ ♣❛ss❡✐♦ ❢❡❝❤❛❞♦ P = (u0, u1, ..., uk−1, u0)

q✉❡ ♥ã♦ ❛tr❛✈❡ss❛ ✉♠❛ ♠❡s♠❛ ❛r❡st❛ ❞✉❛s ✈❡③❡s✳ ❙❡❥❛♠ H₁, ..., Hl ❛s ❝♦♠♣♦♥❡♥t❡s

❝♦♥❡①❛s ♥ã♦ tr✐✈✐❛s ❞❡ G−A✱ ♦♥❞❡A é ♦ ❝♦♥❥✉♥t♦ ❞❛s ❛r❡st❛s ❞❡ P✳ ❙❡ ✉♠ ✈ért✐❝❡

u ❞❡ Hi ♦❝♦rr❡ j ≥0 ✈❡③❡s ❡♠ P✱ ❡♥tã♦

dG(u) = dHi(u) + 2j;

❞❡ ♠♦❞♦ q✉❡dHi(u)é ♣❛r✳ ❈♦♠♦ ✐ss♦ é ✈á❧✐❞♦ ♣❛r t♦❞♦s1≤i≤l ❡u✱ s❡❣✉❡ q✉❡

H1, ..., Hl s❛t✐s❢❛③❡♠ ❛s ❤✐♣ót❡s❡s ❞♦ t❡♦r❡♠❛✳

❈♦♠♦ ❝❛❞❛Hi t❡♠ ♠❡♥♦s ❛r❡st❛s q✉❡G✱ s❡❣✉❡ ❞❛ ❤✐♣ót❡s❡ ❞❡ ✐♥❞✉çã♦ ❛ ❡①✐stê♥✲

❝✐❛ ❞❡ ✉♠ ♣❛ss❡✐♦ ❊✉❧❡r✐❛♥♦Pi ♣❛r❛Hi✳ ❆ ✐❞❡✐❛ ❛❣♦r❛ é ❝♦❧❛r ♦s ♣❛ss❡✐♦s ❊✉❧❡r✐❛♥♦s

P1, ...,Pl ✉♠ ❛ ✉♠ ❛ P✱ ♦❜t❡♥❞♦✱ ❛ss✐♠ ✉♠ ♣❛ss❡✐♦ ❊✉❧❡r✐❛♥♦ ♣❛r❛ G✳ P❛r❛ t❛♥t♦✱

♥♦t❡ q✉❡ P ❡ P1 tê♠ ♣❡❧♦ ♠❡♥♦s ✉♠ ✈ért✐❝❡ ❝♦♠✉♠✱ ❞✐❣❛♠♦s u✳ P♦❞❡♠♦s s✉♣♦r✱

s❡♠ ♣❡r❞❛ ❞❡ ❣❡♥❡r❛❧✐❞❛❞❡✱ q✉❡ P ❡ P1 ❝♦♠❡ç❛♠ ❡ t❡r♠✐♥❛♠ ❡♠ u ✭♠✉❞❛♥❞♦ ♦s

♣♦♥t♦s ❞❡ ♣❛rt✐❞❛ ❞❡ ❛♠❜♦s ❡ss❡s ♣❛ss❡✐♦s✱ s❡ ♥❡❝❡ssár✐♦✮✳ P♦rt❛♥t♦✱ s❡❣✉✐♥❞♦ P ❡

❡♠ s❡❣✉✐❞❛P1✱ ♦❜t❡♠♦s ✉♠ ♣❛ss❡✐♦P ′

1 ❡♠Gq✉❡ ❡♥❣❧♦❜❛ t♦❞❛s ❛s ❛r❡st❛s ❞❡P ❡ ❞❡

P1 ❡ ♥ã♦ ♣❛ss❛ ❞✉❛s ✈❡③❡s ♣♦r ✉♠❛ ♠❡s♠❛ ❛r❡st❛✳ ❇❛st❛ ❛❣♦r❛ r❡♣❡t✐r ♦ ❛r❣✉♠❡♥t♦

❛❝✐♠❛ ♠❛✐sl−1✈❡③❡s✱ ❝♦❧❛♥❞♦P2 ❛P ′

1♣❛r❛ ♦❜t❡r ✉♠ ♣❛ss❡✐♦P ′

2 ❡ ❛ss✐♠ ♣♦r ❞✐❛♥t❡✳

❖ ♣❛ss❡✐♦ P′

l ♦❜t✐❞♦ ♥♦ ✜♥❛❧ é ✉♠ ♣❛ss❡✐♦ ❊✉❧❡r✐❛♥♦ ❡♠ G✳

✷✳✸ ❈✐❝❧♦s ❍❛♠✐❧t♦♥✐❛♥♦s

❊st❛ s❡çã♦ ❢♦✐ ❡❧❛❜♦r❛❞❛ ❛ ♣❛rt✐r ❞❛s s❡❣✉✐♥t❡s r❡❢❡rê♥❝✐❛s ❜✐❜❧✐♦❣rá✜✲ ❝❛s✿ ❬✶❪✱ ❬✷❪ ❡ ❬✸❪✳

❉❡✜♥✐çã♦ ✷✳✼ ❯♠ ❝✐❝❧♦ ❍❛♠✐❧t♦♥✐❛♥♦ ❡♠ ✉♠ ❣r❛❢♦ é ✉♠ ❝✐❝❧♦ q✉❡ ♣❛ss❛ ♣♦r t♦❞♦s ♦s ✈ért✐❝❡s ❞♦ ❣r❛❢♦✳ ❯♠ ❣r❛❢♦ ❝♦♥❡①♦ é ❍❛♠✐❧t♦♥✐❛♥♦ s❡ ❝♦♥t✐✈❡r ✉♠ ❝✐❝❧♦ ❍❛♠✐❧t♦♥✐❛♥♦✳

❖ ♣r♦❜❧❡♠❛ ❞❛ ❡①✐stê♥❝✐❛ ♦✉ ♥ã♦ ❞❡ ❝✐❝❧♦s ❍❛♠✐❧t♦♥✐❛♥♦s ❡♠ ❣r❛❢♦s é ♠✉✐t♦ ♠❛✐s ❞✐❢í❝✐❧ q✉❡ ♦ ♣r♦❜❧❡♠❛ ❝♦rr❡s♣♦♥❞❡♥t❡ ♣❛r❛ ♣❛ss❡✐♦s ❊✉❧❡r✐❛♥♦s❀ ❞❡ ❢❛t♦✱ ❛té ❤♦❥❡ ♥ã♦ ❡①✐st❡ ✉♠ ❝♦♥❥✉♥t♦ ❞❡ ❝♦♥❞✐çõ❡s ♥❡❝❡ssár✐❛s ❡ s✉✜❝✐❡♥t❡s s✐♠♣❧❡s ♣❛r❛ ✉♠ ❣r❛❢♦ s❡r ❍❛♠✐❧t♦♥✐❛♥♦✳ ❖ t❡♦r❡♠❛ ❛ s❡❣✉✐r✱ ❞❡✈✐❞♦ ❛♦ ♠❛t❡♠át✐❝♦ ❤ú♥❣❛r♦ ❞♦ sé❝✉❧♦ ❳❳ ●❛❜r✐❡❧ ❉✐r❛❝✱ ❞á ✉♠❛ ❝♦♥❞✐çã♦ s✉✜❝✐❡♥t❡ s✐♠♣❧❡s ♣❛r❛ ❛ ❡①✐stê♥❝✐❛ ❞❡ ❝✐❝❧♦s ❍❛♠✐❧t♦♥✐❛♥♦s✳

P❛ss❡✐♦s✱ ❈❛♠✐♥❤♦s✱ ❈✐❝❧♦s ❡ ➪r✈♦r❡s ❈❛♣ít✉❧♦ ✷

❚❡♦r❡♠❛ ✷✳✸ ✭❉✐r❛❝✮ ❙❡ G é ✉♠ ❣r❛❢♦ ❝♦♠ n ≥ 3 ✈ért✐❝❡s ❡ t❛❧ q✉❡ δ(G) ≥ n 2✱

❡♥tã♦ G é ❍❛♠✐❧t♦♥✐❛♥♦✳

Pr♦✈❛✳

P❡❧❛ ♣r♦♣♦s✐çã♦ ✷✳✶✱G é ❝♦♥❡①♦✳ ❙❡❥❛ ❛❣♦r❛ P = (u0, ..., uk) ✉♠ ❝❛♠✐♥❤♦ ♠❛①✐✲

♠❛❧ ❡♠ ● ✭✐✳❡✳✱ t❛❧ q✉❡ ♥ã♦ ❤á ♦✉tr♦ ❞❡ ❝♦♠♣r✐♠❡♥t♦ ♠❛✐♦r✮✳ ❙❡ v ❢♦r ✉♠ ✈✐③✐♥❤♦

❞❡u0 q✉❡ ♥ã♦ ♣❡rt❡♥❝❡ ❛ P✱ ♣♦❞❡♠♦s tr♦❝❛r P ♣♦rP1 = (v, u0, ..., uk)✱ ♦❜t❡♥❞♦ ✉♠

❝❛♠✐♥❤♦ ♠❛✐♦r q✉❡P✱ ♦ q✉❡ é ✉♠❛ ❝♦♥tr❛❞✐çã♦✳ P♦rt❛♥t♦✱P ✐♥❝❧✉✐ t♦❞♦s ♦s ✈ért✐❝❡s

❞❡u₀ ❡✱ ❛♥❛❧♦❣❛♠❡♥t❡✱ t♦❞♦s ♦s ✈✐③✐♥❤♦s ❞❡uk✳ ❊♠ ♣❛rt✐❝✉❧❛r✱P t❡♠ ❝♦♠♣r✐♠❡♥t♦

♠❛✐♦r ♦✉ ✐❣✉❛❧ ❛ n 2✳

❆✜r♠❛♠♦s ❛❣♦r❛ q✉❡ ❡①✐st❡ ✉♠ í♥❞✐❝❡ 0 ≤ i ≤ k−1 t❛❧ q✉❡ u0 é ❛❞❥❛❝❡♥t❡ ❛

u_i+1 ❡ u_i é ❛❞❥❛❝❡♥t❡ u_k✳ P❛r❛ t❛♥t♦✱ s❡❥❛♠ A = {0≤ i ≤ k−1;u_i+1 ∈ N_G(u₀)} ❡

B = {0 ≤ i ≤ k −1;ui ∈ NG(uk)}; ❝♦♠♦ t♦❞♦s ♦s ✈ért✐❝❡s ❡♠ NG(u0) ❡ t♦❞♦s ♦s

✈ért✐❝❡s ❡♠ NG(uk) ❡stã♦ ❡♠ P✱ s❡❣✉❡ q✉❡ |A|,|B| ≥ n₂✳ P♦rt❛♥t♦✱

|A∩B|=|A|+|B| − |A∪B| ≥n−k > 0,

✉♠❛ ✈❡③ q✉❡ P é ✉♠ ❝❛♠✐♥❤♦ ❡♠ G❝♦♠ k+ 1 ✈ért✐❝❡s✳

❚♦♠❡✱ ❡♥tã♦✱0≤i≤k−1 ❝♦♠♦ ♥❛ ❛✜r♠❛çã♦✱ ❡ ❝♦♥s✐❞❡r❡ ♦ ❝✐❝❧♦

C = (u₀, u_i+1, u_i+2, ..., uk−1, uk, ui, ui−1, ..., u1, u0).

❆✜r♠❛♠♦s q✉❡C é ❍❛♠✐❧t♦♥✐❛♥♦✳ P♦r ❝♦♥tr❛❞✐çã♦✱ s✉♣♦♥❤❛ q✉❡ ❛❧❣✉♠ ✈ért✐❝❡

❞❡G✱ ❞✐❣❛♠♦sv✱ ♥ã♦ ❡st❡❥❛ ❡♠C✳ ❚♦♠❡ ✉♠ ✈ért✐❝❡w❡♠C✱ ✈✐③✐♥❤♦ ❞❡v✭t❛❧ ✈ért✐❝❡

❡①✐st❡ ♣❡❧♦ ♣r✐♥❝í♣✐♦ ❞❛ ❝❛s❛ ❞♦s ♣♦♠❜♦s✱ ♣♦✐sC t❡♠ ♠❛✐s ❞❡ n

2 ✈ért✐❝❡s ❡v t❡♠ ♣❡❧♦

♠❡♥♦s n

2 ✈✐③✐♥❤♦s✮✳ ❘❡♥♦♠❡❛♥❞♦ ♦s ✈ért✐❝❡s ❞❡ C✱ ♣♦❞❡♠♦s s✉♣♦r q✉❡

C = (v₀, v₁, ..., v_k, v₀),

❝♦♠v₀ =w❀ ♠❛s ❛í✱ P1 = (v, v0, v1, ..., vk) s❡r✐❛ ✉♠ ❝❛♠✐♥❤♦ ♠❛✐♦r q✉❡P ❡♠G✱

✉♠ ❛❜s✉r❞♦✳

✷✳✹ ➪r✈♦r❡s

❊st❛ s❡çã♦ ❢♦✐ ❡❧❛❜♦r❛❞❛ ❛ ♣❛rt✐r ❞❛s s❡❣✉✐♥t❡s r❡❢❡rê♥❝✐❛s ❜✐❜❧✐♦❣rá✜✲ ❝❛s✿ ❬✶❪✱ ❬✷❪ ❡ ❬✸❪✳

❉❡✜♥✐çã♦ ✷✳✽ ❯♠❛ ár✈♦r❡ é ✉♠ ❣r❛❢♦ ❝♦♥❡①♦ ❡ ❛❝í❝❧✐❝♦✳

P❛ss❡✐♦s✱ ❈❛♠✐♥❤♦s✱ ❈✐❝❧♦s ❡ ➪r✈♦r❡s ❈❛♣ít✉❧♦ ✷

❋✐❣✉r❛ ✷✳✷✿ ❯♠❛ ár✈♦r❡✳

❙❡T é ✉♠❛ ár✈♦r❡✱ ✉♠❛ ❢♦❧❤❛ ❞❡ T é ✉♠ ✈ért✐❝❡ ❞❡ ❣r❛✉ ✶ ❡ ✉♠ ♥ó ❞❡ T é ✉♠

✈ért✐❝❡ ❞❡ ❣r❛✉ ♠❛✐♦r q✉❡ ✶✳

▲❡♠❛ ✷✳✷ ❚♦❞❛ ár✈♦r❡ ❝♦♠ ♠❛✐s ❞❡ ✉♠ ✈ért✐❝❡ t❡♠ ♣❡❧♦ ♠❡♥♦s ❞✉❛s ❢♦❧❤❛s✳

Pr♦✈❛✳

❙❡❥❛T ✉♠❛ ár✈♦r❡ ❝♦♠ ♣❡❧♦ ♠❡♥♦s ❞♦✐s ✈ért✐❝❡s ❡ ❝♦♥s✐❞❡r❡ ❡♠ T ✉♠ ❝❛♠✐♥❤♦

C ❝♦♠ ♦ ♠❛✐♦r ❝♦♠♣r✐♠❡♥t♦ ♣♦ssí✈❡❧✳ ❙❡C = (u₁, u₂, ..., un)✱ ❛✜r♠❛♠♦s q✉❡ u1 ❡ un

sã♦ ❢♦❧❤❛s✳ ❉❡ ❢❛t♦✱ s❡ ❢♦ss❡ dT(u1)≥2✱ ❡♥tã♦ u1 s❡r✐❛ ❛❞❥❛❝❡♥t❡ ❛u2 ❡ ❛ ✉♠ ♦✉tr♦

✈ért✐❝❡✱ u₀ ❞✐❣❛♠♦s✳ ❈♦♠♦ ár✈♦r❡s ♥ã♦ ❝♦♥tê♠ ❝✐❝❧♦s✱ ❞❡✈❡ s❡r u₀ 6=u₃, ..., un✳ ❉❛í✱

♦ ❝❛♠✐♥❤♦ (u₀, u₁, u₂, ..., un) t❡r✐❛ ❝♦♠♣r✐♠❡♥t♦ ♠❛✐♦r q✉❡ C✱ ♦ q✉❡ é ✉♠ ❛❜s✉r❞♦✳

▲♦❣♦✱ dT(u1) = 1 ❡✱ ❛♥❛❧♦❣❛♠❡♥t❡✱ dT(un) = 1✳

Pr♦♣♦s✐çã♦ ✷✳✷ ❙❡ T = (V;E) é ✉♠❛ ár✈♦r❡✱ ❡♥tã♦ |E|=|V| −1✳

Pr♦✈❛✳

❋❛ç❛♠♦s ✐♥❞✉çã♦ s♦❜r❡ |V|✳ ■♥✐❝✐❛❧♠❡♥t❡✱ é ❝❧❛r♦ q✉❡ t♦❞❛ ár✈♦r❡ ❝♦♠ ❞♦✐s

✈ért✐❝❡s t❡♠ ❡①❛t❛♠❡♥t❡ ✉♠❛ ❛r❡st❛✳ ❙✉♣♦♥❤❛✱ ❝♦♠♦ ❤✐♣ót❡s❡ ❞❡ ✐♥❞✉çã♦✱ q✉❡ t♦❞❛ ár✈♦r❡ ❝♦♠ n > 1 ✈ért✐❝❡s t❡♠ ❡①❛t❛♠❡♥t❡ n−1 ❛r❡st❛s✱ ❡ ❝♦♥s✐❞❡r❡ ✉♠❛ ár✈♦r❡

T ❝♦♠ n+ 1 ✈ért✐❝❡s✳ P❡❧♦ ❧❡♠❛ ❛♥t❡r✐♦r✱ T t❡♠ ✉♠❛ ❢♦❧❤❛ u❀ ♣♦rt❛♥t♦✱ T −u é

✉♠❛ ár✈♦r❡ ❝♦♠ n ✈ért✐❝❡s✱ ❡ ❛ ❤✐♣ót❡s❡ ❞❡ ✐♥❞✉çã♦ ❣❛r❛♥t❡ q✉❡ T −u t❡♠ n−1

❛r❡st❛s✳ ▼❛s ❝♦♠♦ T t❡♠ ❡①❛t❛♠❡♥t❡ ✉♠❛ ❛r❡st❛ ❛ ♠❛✐s q✉❡ T −u✱ s❡❣✉❡ q✉❡ T

t❡♠ ❡①❛t❛♠❡♥t❡ n ❛r❡st❛s✳

❈♦r♦❧ár✐♦ ✷✳✶ ❙❡ G= (V;E) é ✉♠ ❣r❛❢♦ ❝♦♥❡①♦✱ ❡♥tã♦ G ❝♦♥té♠ ♣❡❧♦ ♠❡♥♦s ✉♠

❝✐❝❧♦ s❡ ❡ só s❡ |E| ≥ |V|✳

Pr♦✈❛✳

❙✉♣♦♥❤❛ ✐♥✐❝✐❛❧♠❡♥t❡ q✉❡ G ❝♦♥té♠ ♣❡❧♦ ♠❡♥♦s ✉♠ ❝✐❝❧♦✱ ❡ ♠♦str❡♠♦s q✉❡

|E| ≥ |V| ♣♦r ✐♥❞✉çã♦ s♦❜r❡ |E|✳ ❙❡ ǫ é ✉♠❛ ❛r❡st❛ ❞❡ ✉♠ ❝✐❝❧♦ ❡♠ G✱ ❡♥tã♦ G−ǫ

❛✐♥❞❛ é ❝♦♥❡①♦ ❡ t❡♠ |V|✈ért✐❝❡s ❡ |E| −1❛r❡st❛s✳ ❍á✱ ♣♦✐s✱ ❞✉❛s ♣♦ss✐❜✐❧✐❞❛❞❡s✿

P❛ss❡✐♦s✱ ❈❛♠✐♥❤♦s✱ ❈✐❝❧♦s ❡ ➪r✈♦r❡s ❈❛♣ít✉❧♦ ✷

• G−ǫ ❛✐♥❞❛ ❝♦♥té♠ ♣❡❧♦ ♠❡♥♦s ✉♠ ❝✐❝❧♦✿ s❡❣✉❡ ❞❛ ❤✐♣ót❡s❡ ❞❡ ✐♥❞✉çã♦ q✉❡

|E| −1≥ |V|❀

• G−ǫ é ❛❝í❝❧✐❝♦✿ ❝♦♠♦ ❛✐♥❞❛ é ❝♦♥❡①♦✱ G−ǫ é ✉♠❛ ár✈♦r❡✱ ❡ ❛ ♣r♦♣♦s✐çã♦

❛♥t❡r✐♦r ❣❛r❛♥t❡ q✉❡ |V|= (|E| −1) + 1 =|E|✳

❊♠ q✉❛❧q✉❡r ❝❛s♦✱|E| ≥ |V|✳

❘❡❝✐♣r♦❝❛♠❡♥t❡✱ s❡G❢♦r ❛❝í❝❧✐❝♦✱ ❡♥tã♦Gé ✉♠❛ ár✈♦r❡✱ ❡ ❛ ♣r♦♣♦s✐çã♦ ❛♥t❡r✐♦r

❣❛r❛♥t❡ q✉❡ |E|=|V| −1<|V|✳

❈♦r♦❧ár✐♦ ✷✳✷ ❙❡❥❛ T = (V;E) ✉♠ ❣r❛❢♦ ❝♦♥❡①♦✳ ❙❡ |E|=|V| −1✱ ❡♥tã♦T é ✉♠❛

ár✈♦r❡✳

Pr♦✈❛✳

❙❡T ❝♦♥t✐✈❡ss❡ ♣❡❧♦ ♠❡♥♦s ✉♠ ❝✐❝❧♦✱ s❡❣✉✐r✐❛ ❞♦ ❝♦r♦❧ár✐♦ ❛♥t❡r✐♦r q✉❡|E| ≥ |V|✱

✉♠❛ ❝♦♥tr❛❞✐çã♦✳ ▲♦❣♦✱ T é ❝♦♥❡①♦ ❡ ❛❝í❝❧✐❝♦✱ ❞♦♥❞❡ ✉♠❛ ár✈♦r❡✳

❈❛♣ít✉❧♦ ✸

P❧❛♥❛r✐❞❛❞❡

✸✳✶ ❘❡♣r❡s❡♥t❛çõ❡s ❡ ❈r✉③❛♠❡♥t♦s

❊st❛ s❡çã♦ ❢♦✐ ❡❧❛❜♦r❛❞❛ ❛ ♣❛rt✐r ❞❛ s❡❣✉✐♥t❡ r❡❢❡rê♥❝✐❛ ❜✐❜❧✐♦❣rá✜❝❛✿ ❬✸❪✳

❖s ❞♦✐s ❞✐❛❣r❛♠❛s ♥❛ ✜❣✉r❛ ✸✳✶ r❡♣r❡s❡♥t❛♠ ♦ ♠❡s♠♦ ❣r❛❢♦✱ ✉♠ K4 ❝♦♠ ✈ér✲

t✐❝❡s a✱ b✱ c ❡ d✳ ❈♦♠♦ ♦s ❞✐❛❣r❛♠❛s sã♦ ❜❛st❛♥t❡ ❞✐❢❡r❡♥t❡s✿ ♥❛ ✈❡rsã♦ K₄(1)✱

❛s ❛r❡st❛s ac ❡ bd s❡ ❝r✉③❛♠❀ ❡♠ K₄(2) ♥ã♦ ❤á ❝r✉③❛♠❡♥t♦s✳ ❉❡✈❡♠♦s ♥♦s r❡❢❡r✐r

❛♦s ❞♦✐s ❞✐❛❣r❛♠❛s ❝♦♠♦ ❞✐❢❡r❡♥t❡s r❡♣r❡s❡♥t❛çõ❡s ❞♦ ❣r❛❢♦ ♥♦ ♣❧❛♥♦✳ ❖ ♥ú♠❡r♦ ❞❡ ❝r✉③❛♠❡♥t♦s ❞❡ ✉♠❛ r❡♣r❡s❡♥t❛çã♦ é ♦ ♥ú♠❡r♦ ❞❡ ❞✐❢❡r❡♥t❡s ♣❛r❡s ❞❡ ❛r❡st❛s q✉❡ s❡ ❝r✉③❛♠❀ ♦ ♥ú♠❡r♦ ❞❡ ❝r✉③❛♠❡♥t♦s v(G) ❞❡ ✉♠ ❣r❛❢♦ G é ♦ ♥ú♠❡r♦ ♠í♥✐♠♦

❞❡ ❝r✉③❛♠❡♥t♦s ❡♠ q✉❛❧q✉❡r r❡♣r❡s❡♥t❛çã♦ ❞❡ G✳ ❯♠❛ r❡♣r❡s❡♥t❛çã♦ é ❝❤❛♠❛❞❛

♣❧❛♥❛r s❡ ♥ã♦ ❤♦✉✈❡r ❝r✉③❛♠❡♥t♦s✱ ❡ ✉♠ ❣r❛❢♦ ♣❧❛♥❛r é ✉♠ ❣r❛❢♦ q✉❡ t❡♠ ✉♠❛ r❡♣r❡s❡♥t❛çã♦ ♣❧❛♥❛r✳ ❊♠ ♦✉tr❛s ♣❛❧❛✈r❛s✱ ✉♠ ❣r❛❢♦ ♣❧❛♥❛rGé ❛q✉❡❧❡ ♣❛r❛ ♦ q✉❛❧ v(G) = 0✳ ❆ ✜❣✉r❛ ✸✳✶ ♠♦str❛ q✉❡ v(K4) = 0✳

❋✐❣✉r❛ ✸✳✶✿ ❉✉❛s r❡♣r❡s❡♥t❛çõ❡s ❞❡ K4✳

❍á ♠✉✐t❛s ❛♣❧✐❝❛çõ❡s ❞❡ ♥ú♠❡r♦s ❞❡ ❝r✉③❛♠❡♥t♦s✳ ❯♠ ✉s♦ ♣r❡❝♦❝❡ ❢♦✐ ♥♦ ♣r♦❥❡t♦ ♠❡tr♦❢❡rr♦✈✐ár✐♦s✱ ♦♥❞❡ é ✐♥❝♦♥✈❡♥✐❡♥t❡ t❡r ❞✐❢❡r❡♥t❡s ❧✐♥❤❛s q✉❡ s❡ ❝r✉③❛♠✱ é ♠❡✲

P❧❛♥❛r✐❞❛❞❡ ❈❛♣ít✉❧♦ ✸

❧❤♦r t❡r ♠❛✐s ♣✐st❛✱ ❡♠ ✈❡③ ❞❡ ❝r✉③❛♠❡♥t♦s ❡①tr❛s✳ ❯♠❛ ❡①t❡♥sã♦ ó❜✈✐❛ ❞❡ss❛ ✐❞❡✐❛ é ♦ ♣r♦❥❡t♦ ❛✉t♦❡str❛❞❛✳ ❊♠ ✉♠ ❝r✉③❛♠❡♥t♦ ❝♦♠♣❧❡①♦✱ ♠❡♥♦s ❝r✉③❛♠❡♥t♦s s✐❣♥✐✲ ✜❝❛ ♠❡♥♦s ✈✐❛❞✉t♦s ❝❛r♦s✳ ▼❛✐s r❡❝❡♥t❡♠❡♥t❡✱ ♣❡q✉❡♥♦s ♥ú♠❡r♦s ❞❡ ❝r✉③❛♠❡♥t♦s r❡✈❡❧❛r❛♠✲s❡ ✐♠♣♦rt❛♥t❡s ♥♦ ❞❡s❡♥❤♦ ❞❡ ❝❤✐♣s ❱▲❙■❀ s❡ ❞✉❛s ♣❛rt❡s ❞❡ ✉♠ ❝✐r❝✉✐t♦ ♥ã♦ sã♦ ♣❛r❛ s❡r ❧✐❣❛❞♦s ❡❧❡tr✐❝❛♠❡♥t❡s✱ ♠❛s ❡❧❡s s❡ ❝r✉③❛♠✱ é ♥❡❝❡ssár✐♦ ✉♠ ♣r♦❝❡ss♦ ❞❡ ✐s♦❧❛♠❡♥t♦ ❞✐s♣❡♥❞✐♦s♦✳

❉❡✜♥✐çã♦ ✸✳✶ ❙✉♣♦♥❤❛♠♦s q✉❡ G é ♣❧❛♥❛r✳ ❙❡ ✉♠ ♥♦✈♦ ❣r❛❢♦ ❢♦✐ ❝♦♥str✉í❞♦ ❛tr❛✲

✈és ❞❛ ✐♥s❡rçã♦ ❞❡ ✉♠ ♥♦✈♦ ✈ért✐❝❡ ❞❡ ❣r❛✉ ✷ ♥♦ ♠❡✐♦ ❞❡ ✉♠❛ ❛r❡st❛ ✭❞✐✈✐❞✐♥❞♦ ✉♠❛ ❛r❡st❛✮✱ ♦✉ ♣♦r ❡❧✐♠✐♥❛çã♦ ❞❡ ✉♠ ✈ért✐❝❡ ❞❡ ❣r❛✉ ✷ ❡ ❥✉♥t❛♥❞♦ ♦s ❞♦✐s ✈ért✐❝❡s ❛❞❥❛❝❡♥t❡s ❛ ❡❧❡✱ ♦ ♥♦✈♦ ❣r❛❢♦ t❛♠❜é♠ s❡rá ♣❧❛♥❛r✳

●r❛❢♦s q✉❡ ♣♦❞❡♠ s❡r ♦❜t✐❞♦s ❛ ♣❛rt✐r ❞❡ ♦✉tr♦ ❞❡st❛ ♠❛♥❡✐r❛ sã♦ ❝❤❛♠❛❞♦s ❤♦♠❡♦♠♦r❢♦s✳

❯♠❛ ✈❡③ q✉❡ K5 ❡ K3,3 ♥ã♦ sã♦ ♣❧❛♥❛r❡s✱ s❡❣✉❡✲s❡ q✉❡ ✉♠ ❣r❛❢♦ q✉❡ t❡♠ K5

♦✉ K_3,3 ❝♦♠♦ ✉♠ s✉❜❣r❛❢♦ ♥ã♦ ♣♦❞❡ s❡r ♣❧❛♥❛r✳ ❆❧é♠ ❞✐ss♦✱ ✉♠ ❣r❛❢♦ q✉❡ é ❤♦♠❡✲

♦♠♦r❢♦ ❛ ✉♠ ❣r❛❢♦ ❝♦♠ s✉❜❣r❛❢♦ ❤♦♠❡♦♠♦r❢♦ ❛ K₅ ♦✉ K_3,3 ♥ã♦ ♣♦❞❡ s❡r ♣❧❛♥❛r✳

◆❛ ✈❡r❞❛❞❡✱ ❑✉r❛t♦✇s❦✐ ♣r♦✈♦✉ q✉❡ ❡st❛ ❝♦♥❞✐çã♦ ♥❡❝❡ssár✐❛ ♣❛r❛ ❛ ♣❧❛♥❛r✐❞❛❞❡ t❛♠❜é♠ é s✉✜❝✐❡♥t❡✳

❚❡♦r❡♠❛ ✸✳✶ ✭❑✉r❛t♦✇s❦✐✮ G é ♣❧❛♥❛r s❡ ❡ s♦♠❡♥t❡ s❡ G é ❤♦♠❡♦♠♦r❢♦ ❛ ✉♠

❣r❛❢♦ q✉❡ ♥ã♦ ❝♦♥té♠ s✉❜❣r❛❢♦ ❤♦♠❡♦♠♦r❢♦ ❛ K₅ ♦✉ K_3,3✳

❊st❡ t❡♦r❡♠❛ ❢♦✐ ❛♣r❡s❡♥t❛❞♦ ♣❡❧♦ ♠❛t❡♠át✐❝♦ ♣♦❧♦♥ês ❑✉r❛t♦✇s❦✐ ❡♠ ✶✾✸✵✳ ❆ s✉❛ ❞❡♠♦♥str❛çã♦ é ❜❛st❛♥t❡ ❧♦♥❣❛ ❡ ❝♦♠♣❧❡①❛✳ ❚❡♠ ❛ ✈✐rt✉❞❡ ❞❡ ❢♦r♥❡❝❡r ✉♠❛ ❝♦♥❞✐çã♦ ♥❡❝❡ssár✐❛ ❡ s✉✜❝✐❡♥t❡ ♣❛r❛ ❛ ♣❧❛♥❛r✐❞❛❞❡ ❞❡ ✉♠ ❣r❛❢♦ q✉❡ ♥ã♦ é ❞❡♣❡♥❞❡♥t❡ ❞❛ s✉❛ r❡♣r❡s❡♥t❛çã♦ ❣rá✜❝❛✳ ◆ã♦ ❡①✐st❡ ♥❡♥❤✉♠ ❛❧❣♦r✐t♠♦ ♥❡❧❡ ❜❛s❡❛❞♦ ♣❛r❛ t❡st❛r ❛ ♣❧❛♥❛r✐❞❛❞❡✴♥ã♦ ♣❧❛♥❛r✐❞❛❞❡ ❞❡ ✉♠ ❣r❛❢♦✳ ◆❛ ✈❡r❞❛❞❡✱ t❡st❛r t♦❞♦s ♦s s✉❜❣r❛❢♦s ❡ ✈❡r✐✜❝❛r s❡ sã♦ ❤♦♠❡♦♠♦r❢♦s ❛ K5 ♦✉ ❛ K3,3 é ✉♠❛ t❛r❡❢❛ ár❞✉❛✳

❊①❡♠♣❧♦ ✸✳✶ ❈♦♠♦ ❛♣❧✐❝❛çã♦ ♠♦str❛♠♦s ♥❛ ✜❣✉r❛ ✸✳✷ q✉❡ ♦ ❣r❛❢♦ ❞❡ P❡t❡rs❡♥ ♥ã♦ é ♣❧❛♥❛r✳

❋✐❣✉r❛ ✸✳✷✿ ●r❛❢♦ ❞❡ P❡t❡rs❡♥ ❡ ❤♦♠❡♦♠♦r❢♦s✳

P❧❛♥❛r✐❞❛❞❡ ❈❛♣ít✉❧♦ ✸

✸✳✷ ❋ór♠✉❧❛ ❞❡ ❊✉❧❡r

❊st❛ s❡çã♦ ❢♦✐ ❡❧❛❜♦r❛❞❛ ❛ ♣❛rt✐r ❞❛ s❡❣✉✐♥t❡ r❡❢❡rê♥❝✐❛ ❜✐❜❧✐♦❣rá✜❝❛✿ ❬✸❪✳

❊♠ ❝❛❞❛ ♣❧❛♥♦ ❞❡ r❡♣r❡s❡♥t❛çã♦ ❞❡ ✉♠ ❣r❛❢♦ ❝♦♥❡①♦✱ ♦ ♣❧❛♥♦ é ❞✐✈✐❞✐❞♦ ❡♠ r❡❣✐õ❡s ❝❤❛♠❛❞❛s ❢❛❝❡s✿ ✉♠❛ ❢❛❝❡ é ✉♠❛ ár❡❛ ❞♦ ♣❧❛♥♦ t♦t❛❧♠❡♥t❡ r♦❞❡❛❞❛ ♣♦r ❛r❡st❛s ❞♦ ❣r❛❢♦✱ q✉❡ ♥ã♦ ❝♦♥té♠ q✉❛❧q✉❡r ❛r❡st❛✳ ➱ ❝♦♥✈❡♥✐❡♥t❡ ❞❡✜♥✐r ✉♠❛ ❢❛❝❡ ❡①t❡r✐♦r✱ ❝♦rr❡s♣♦♥❞❡♥t❡ ❛♦ ♣❧❛♥♦ ❞❡ r❡♣r❡s❡♥t❛çã♦ ❡①t❡r✐♦r✳ P♦r ❡①❡♠♣❧♦✱ ❛ ❢❛❝❡ ❡①t❡r✐♦r éabyz ♥❛ ❋✐❣✉r❛ ✸✳✷✭❛✮✱ ❡abxyz ♥❛ ❋✐❣✉r❛ ✸✳✷✭❜✮✳ ❖ ❝✐❝❧♦ abxyz ♥ã♦ é ✉♠❛

❢❛❝❡ ♥❛ ❋✐❣✉r❛ ✸✳✷✭❛✮✱ ♣♦✐s ❡❧❡ ❝♦♥té♠ ❛ ❛r❡st❛ bz✳

❋✐❣✉r❛ ✸✳✸✿ ❉✐❢❡r❡♥t❡s ❢❛❝❡s ❡♠ ❞✐❢❡r❡♥t❡s r❡♣r❡s❡♥t❛çõ❡s✳

❚❡♦r❡♠❛ ✸✳✷ ✭❋ór♠✉❧❛ ❞❡ ❊✉❧❡r✮ ❙✉♣♦♥❤❛ q✉❡ ✉♠❛ r❡♣r❡s❡♥t❛çã♦ ♣❧❛♥❛ ❞♦ ❣r❛❢♦ ❝♦♥❡①♦ G t❡♠ v ✈ért✐❝❡s✱ e ❛r❡st❛s ❡ f ❢❛❝❡s✳ ❊♥tã♦✿

v−e+f = 2✳ (3.1)

Pr♦✈❛✳

P♦r ✐♥❞✉çã♦ s♦❜r❡e✳ ➱ ❢á❝✐❧ ✈❡r q✉❡ ♦ t❡♦r❡♠❛ é ✈á❧✐❞♦ q✉❛♥❞♦ e é ♣❡q✉❡♥♦✿ s❡ e = 0✱ ❡♥tã♦ ♦ ❣r❛❢♦ ❞❡✈❡ s❡r K1✱ q✉❡ t❡♠ v = 1 ❡ f = 1❀ s❡ e = 1 ♦✉ e = 2✱ ♥ós

t❡♠♦s ✉♠ ❝❛♠✐♥❤♦ ❝♦♠e+ 1✈ért✐❝❡s ❡ ✉♠❛ ❢❛❝❡✳ ❆❣♦r❛ ❛ss✉♠✐♠♦s ♦ t❡♦r❡♠❛ ❝♦♠♦

✈❡r❞❛❞❡✐r♦ ♣❛r❛ t♦❞♦s ♦s ❣r❛❢♦s ❝♦♠ E ♦✉ ♠❡♥♦s ❛r❡st❛s✱ ❡ s✉♣♦♥❤❛ e=E+ 1✳

❙❡G é ✉♠❛ ár✈♦r❡✱ ❡♥tã♦ v =e+ 1 ❡ f = 1✱ ❧♦❣♦ v−e+f =e+ 1−e+ 1 = 2✳

❈❛s♦ ❝♦♥trár✐♦ G ❝♦♥té♠ ✉♠ ❝✐❝❧♦✳ ❙❡❧❡❝✐♦♥❡ ✉♠❛ ❛r❡st❛ q✉❡ s❡ ❡♥❝♦♥tr❛ ♥❡st❡

❝✐❝❧♦✳ ❆ ❛r❡st❛ s❡ ❡♥❝♦♥tr❛rá s❡♣❛r❛♥❞♦ ❞✉❛s ❢❛❝❡s ✭✉♠❛ ♣♦ss✐✈❡❧♠❡♥t❡ ♣♦❞❡ s❡r ❛ ❢❛❝❡ ❡①t❡r✐♦r✮✳ ❙❡ ❢♦r ❡❧✐♠✐♥❛❞♦✱ ♦❜té♠✲s❡ ✉♠ ❣r❛❢♦ ❝♦♠ ✉♠❛ ❛r❡st❛ ❛ ♠❡♥♦s ❡ ♠❡♥♦s ✉♠❛ ❢❛❝❡ ❞♦ q✉❡ ♦ ♦r✐❣✐♥❛❧✳ ❙❡ ❤á E ❛r❡st❛s ❛ss✐♠✱ ♣♦r ✐♥❞✉çã♦✱ ❛ ❡q✉❛çã♦

✭✸✳✶✮ é s❛t✐s❢❡✐t❛❀ ❡♠ t❡r♠♦s ❞♦ ❣r❛❢♦ ♦r✐❣✐♥❛❧✱

v−(e−1) + (f−1) = 2✱