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❍❛r❞♥❡ss ♦❢ ❈♦♠♣✉t✐♥❣ t❤❡ ❇✐❝❧✐q✉❡ P❛rt✐t✐♦♥ ◆✉♠❜❡r ❆❧❡①❛♥❞❡r ❘✳ ❇❧♦❝❦ ❏✉❧② ✶✼✱ ✷✵✶✼ ✶ ✴ ✶✻

■♥tr♦❞✉❝t✐♦♥ ❇✐❝❧✐q✉❡ P❛rt✐t✐♦♥ ◆✉♠❜❡r ✭✐♥ s❤♦rt✱ bp ✮ ◮ ▼✐♥✐♠✉♠ ♥✉♠❜❡r ♦❢ ❜✐❝❧✐q✉❡s ♥❡❡❞❡❞ t♦ ♣❛rt✐t✐♦♥ t❤❡ ❡❞❣❡s ♦❢ ❛ ❣r❛♣❤ G ✱ ❞❡♥♦t❡❞ bp ( G ) ◮ ◆♦t❡ t❤❛t G ❝❛♥ ❜❡ ❛♥② ❣r❛♣❤ ◮ ❇✐❝❧✐q✉❡s ❛r❡ ❝♦♠♣❧❡t❡ ❜✐♣❛rt✐t❡ ❣r❛♣❤s ✱ ❞❡♥♦t❡❞ K n,m ✷ ✴ ✶✻

■♥tr♦❞✉❝t✐♦♥ ✭❊①❛♠♣❧❡s✮ = + + K 1 , 3 K 1 , 2 K 1 , 1 K 4 = + K 2 , 2 K 2 , 1 K 1 , 1 ✸ ✴ ✶✻

■♥tr♦❞✉❝t✐♦♥ ●r❛❤❛♠ ❛♥❞ P♦❧❧❛❦ ✐♥tr♦❞✉❝❡❞ t❤❡ ❜✐❝❧✐q✉❡ ♣❛rt✐t✐♦♥ ♥✉♠❜❡r ✐♥ ✶✾✼✷ ✐♥ t❤❡ ❝♦♥t❡①t ♦❢ ♥❡t✇♦r❦ ❛❞❞r❡ss✐♥❣ ❛♥❞ ❣r❛♣❤ st♦r❛❣❡ ♣r♦❜❧❡♠s ❬●P✼✶✱ ●P✼✷❪ ■♥tr♦❞✉❝❡❞ ❛♥ ❡①tr❡♠❡❧② ♣r♦❧✐✜❝ r❡s❡❛r❝❤ ❛r❡❛ ✐♥ ▼❛t❤❡♠❛t✐❝s ●r❛❤❛♠✲P♦❧❧❛❦ ❚❤❡♦r❡♠✿ bp ( K n ) = ( n − 1) ✳ ❆❧❧ ♣r♦♦❢s ❛r❡ ❛❧❣❡❜r❛✐❝ ❛♥❞ ♥♦ ♣✉r❡❧② ❝♦♠❜✐♥❛t♦r✐❛❧ ♣r♦♦❢ ✐s ❦♥♦✇♥ ❬●P✼✷✱ ❚✈❡✽✷✱ P❡❝✽✹✱ ❱✐s✵✽✱ ❱✐s✶✸❪ ◮ ❙❤♦✇❡❞ bp ( G ) � max { n + ( A ( G )) , n − ( A ( G )) } ❬❲✐ts❡♥❤❛✉s❡♥✱ ✶✾✽✵s❪ ◮ ❑♥♦✇♥ t❤❛t n − ( A ( K n )) = ( n − 1) ◮ ❈❛♥ ♣❛rt✐t✐♦♥ K n ✐♥t♦ ( n − 1) st❛rs ⋆ ❆ st❛r ✐s ❛ ❜✐❝❧✐q✉❡ ♦❢ t❤❡ ❢♦r♠ K 1 ,i ❢♦r s♦♠❡ ♣♦s✐t✐✈❡ ✐♥t❡❣❡r i ✹ ✴ ✶✻

❆♣♣❧✐❝❛t✐♦♥s ♦❢ ❇✐❝❧✐q✉❡ P❛rt✐t✐♦♥ ●r❛❤❛♠ ❛♥❞ P♦❧❧❛❦ t❤❛t ❛ ♣r♦❜❧❡♠ ♦♥ ❧♦♦♣ s✇✐t❝❤✐♥❣ ✐♥ ♥❡t✇♦r❦✐♥❣ ✐s ❡q✉✐✈❛❧❡♥t t♦ ♣❛rt✐t✐♦♥✐♥❣ ❛ ♠✉❧t✐❣r❛♣❤✱ ②✐❡❧❞✐♥❣ t❤❡✐r ❝❡❧❡❜r❛t❡❞ r❡s✉❧t ❬●P✼✶✱ ●P✼✷✱ ❚❛✐✶✸❪ ❍❛s ❛♣♣❧✐❝❛t✐♦♥s ❢♦r ♣❡r❢❡❝t ❤❛s❤✐♥❣s ❬❚❛✐✶✸❪ ■◆P❯❚✿ n, r, k ∈ N ✇✐t❤ k � r � n ▼■◆■▼■❩❊✿ s✐③❡ ♦❢ F := { f i : [ n ] → [ r ] } ❈❖◆❙❚❘❆■◆❚✿ ∀ K ⊆ [ n ] ✇✐t❤ | K | = k ✱ ∃ i s✉❝❤ t❤❛t f i | K ✐s ✐♥❥❡❝t✐✈❡ ◮ ❆s❦✐♥❣ ❢♦r ✉♥✐q✉❡ i ❛♥❞ r = k = 2 ❛s❦s ❢♦r bp ( K n ) ❈♦♥♥❡❝t✐♦♥s t♦ t❤❡ ♥♦♥❞❡t❡r♠✐s♥✐st✐❝ st❛t❡ ❝♦♠♣❧❡①✐t② ♦❢ ✜♥✐t❡ ❛✉t♦♠❛t❛✱ ♥❛♠❡❧② ✉s❡❞ ❛s ❛ ❧♦✇❡r ❜♦✉♥❞ ♠❡t❤♦❞ ❬●❍✵✻❪ P❧❛② ❛ r♦❧❧ ✐♥ ❛♥❛❧②s✐s ♦❢ ❍▲❆ r❡❛❝t✐♦♥ ♠❛tr✐❝❡s ✉s❡❞ ✐♥ ❜✐♦❧♦❣② ❬◆▼❲❆✼✽❪ ✺ ✴ ✶✻

❆♣♣❧✐❝❛t✐♦♥s ♦❢ ❇✐❝❧✐q✉❡ P❛rt✐t✐♦♥ ❆ r❡❧❛①❛t✐♦♥✿ bp t ( G ) ✱ ❛ ❝♦✈❡r✐♥❣ ♦❢ ❡❞❣❡s ✇✐t❤ ❛t ♠♦st t ✲❜✐❝❧✐q✉❡s ◮ ❊①❛♠✐♥❡❞ ❜② ◆♦❣❛ ❆❧♦♥ ❬❆❧♦✾✼❪ ◮ ❙❤♦✇❡❞ t❤❛t ✇✐t❤ G = K n ✱ bp t ( G ) ✐s ❡q✉✐✈❛❧❡♥t t♦ ✜♥❞✐♥❣ t❤❡ ♠❛① ♥✉♠❜❡r ♦❢ ❜♦①❡s ✐♥ R n t❤❛t ❛r❡ t ✲♥❡✐❣❤❜♦r❧② ◮ ❆❧s♦ s❤♦✇❡❞ t❤❛t bp t ( K n ) � Θ � tn 1 /t � ❲②♥❡r✬s ❝♦♠♠♦♥ ✐♥❢♦r♠❛t✐♦♥ J ( R A , R B ) ✿ ◮ ▼✐♥✐♠✉♠ ✭❛♠♦✉♥t ♦❢✮ ❧❡❛❦❛❣❡ t❤❛t ❦✐❧❧s t❤❡ ♣♦ss✐❜✐❧✐t② ♦❢ ❦❡② ❛❣r❡❡♠❡♥t ◮ min H ( L ) s✉❝❤ t❤❛t I ( R A , R B | L ) = 0 ◮ ❇✐❝❧✐q✉❡s ❛r❡ ✉s❡❧❡ss ❢♦r ❑❆ ✭❜❡❝❛✉s❡ ✵ ♠✉t✉❛❧ ✐♥❢♦r♠❛t✐♦♥✮ ◮ ❘♦✉❣❤❧② ❝♦rr❡s♣♦♥❞s t♦ t❤❡ ❜✐❝❧✐q✉❡ ♣❛rt✐t✐♦♥ ♥✉♠❜❡r ❈❧♦s❡ ❝♦♥♥❡❝t✐♦♥s t♦ ❝♦♠♠✉♥✐❝❛t✐♦♥ ❝♦♠♣❧❡①✐t② ❛♥❞ ❝✐r❝✉✐t ❧♦✇❡r ❜♦✉♥❞s ❬❍❙✶✷✱ ◆❲✾✺✱ ❘❛③✾✷❪ ✻ ✴ ✶✻

❋✐♥❞✐♥❣ bp ✐s ❍❛r❞ ❙✉♣♣♦s❡ ✇❡ ❛r❡ ❣✐✈❡♥ ❛ ✭❜✐♣❛rt✐t❡✮ ❣r❛♣❤ G ❉♦❡s t❤❡r❡ ❡①✐st ❛ ❜✐❝❧✐q✉❡ ♣❛rt✐t✐♦♥ ♦❢ G ♦❢ s✐③❡ k ❄ ❚❤❡ ♣r♦❜❧❡♠ ✐s ◆P✲❈♦♠♣❧❡t❡ ❢♦r ❜♦t❤ ❜✐♣❛rt✐t❡ ❛♥❞ ❣❡♥❡r❛❧ ❣r❛♣❤s ❬❖r❧✼✼✱ ❈✐♦✵✺❪ ❲❡ s❤♦✇ ❛ ♣r♦♦❢ ❢♦r ❜✐♣❛rt✐t❡ ❣r❛♣❤s ❛♥❞ ❛♥♦t❤❡r ❢♦r ❣❡♥❡r❛❧ ❣r❛♣❤s✳ Pr♦♦❢ ❢♦r ❜✐♣❛rt✐t❡ ❣r❛♣❤s ✐s ❛ r❡❞✉❝t✐♦♥ ❢r♦♠ t❤❡ ✈❡rt❡① ❝❧✐q✉❡ ♣r♦❜❧❡♠ ●■❱❊◆✿ ●r❛♣❤ G ✇✐t❤ V ( G ) = { v 1 , . . . , v n } ❉❊❚❊❘▼■◆❊✿ ❋❡✇❡st ♥✉♠❜❡r ♦❢ ❝❧✐q✉❡s ✇❤✐❝❤ ✐♥❝❧✉❞❡ ❛❧❧ ♦❢ V ( G ) Pr♦♦❢ ❢♦r ❣❡♥❡r❛❧ ❣r❛♣❤s ✐s ❛ r❡❞✉❝t✐♦♥ ❢r♦♠ t❤❡ ✈❡rt❡① ❝♦✈❡r ♣r♦❜❧❡♠ ✼ ✴ ✶✻

Pr♦♦❢ ♦❢ ◆P✲❈♦♠♣❧❡t❡♥❡ss ♦❢ bp ❢♦r ❜✐♣❛rt✐t❡ ❣r❛♣❤s ❙✉♣♣♦s❡ ✇❡ ❛r❡ ❣✐✈❡♥ ❛ G ✐♥ t❤❡ ✐♥st❛♥❝❡ ♦❢ t❤❡ ✈❡rt❡① ❝❧✐q✉❡ ♣r♦❜❧❡♠ ❞❡s❝r✐❜❡❞ ❜❡❢♦r❡ ❛♥❞ ✇❡ ✇❛♥t t♦ ❛♥s✇❡r t❤❡ ❢♦❧❧♦✇✐♥❣ q✉❡st✐♦♥ ●■❱❊◆✿ ❇✐♣❛rt✐t❡ ❣r❛♣❤ G ❉❊❚❊❘▼■◆❊✿ ❋❡✇❡st ♥✉♠❜❡r ♦❢ ❜✐❝❧✐q✉❡s ✇❤✐❝❤ ♣❛rt✐t✐♦♥ ❛ s✉❜s❡t H ⊆ E ( G ) ❈♦♥str✉❝t G ′ = ( L, R, E ′ ) ✇✐t❤ L = { x 1 , . . . , x n } ✱ R = { y 1 , . . . , y n } ❛♥❞ E ′ = { ( x i , y i ): ∀ i } ∪ { ( x i , y j ): ( v i , v j ) ∈ E ( G ) } ▲❡t H ′ = { ( x i , y j ): i = 1 , . . . , n } ❜❡ t❤❡ s❡t ♦❢ ❡❞❣❡s t♦ ❜❡ ❝♦✈❡r❡❞✳ ❆♥② ❝❧✐q✉❡ C ✐♥ G ✇❤✐❝❤ ✐♥❝❧✉❞❡s v i ✐♥❞✉❝❡s ❛ ❜✐❝❧✐q✉❡ ✐♥ G ′ ✇❤✐❝❤ ✐♥❝❧✉❞❡s t❤❡ ❡❞❣❡ ( x i , y i ) ✳ ■❢ C ′ ✐s ❛ ❜✐❝❧✐q✉❡ ♦❢ G ′ ✇❤✐❝❤ ✐♥❝❧✉❞❡s ❡❞❣❡s ( x j 1 , y j 1 ) , ( x j 2 , y j 2 ) , . . . , ( x j k , y j k ) ✱ t❤❡♥ ❜② ❝♦♥str✉❝t✐♦♥ ✐t ♠✉st ❜❡ t❤❡ ❝❛s❡ t❤❛t { v j 1 , . . . , v j k } ✐s ❛ ❝❧✐q✉❡ ✐♥ G ✳ ❙♦ t❤❡ ♠✐♥✐♠✉♠ ♥✉♠❜❡r ♦❢ ❝❧✐q✉❡s t❤❛t ❝♦✈❡r ❛❧❧ ✈❡rt✐❝❡s ✐♥ G ✐s ❡q✉❛❧ t♦ t❤❡ ♠✐♥✐♠✉♠ ♥✉♠❜❡r ♦❢ ❜✐❝❧✐q✉❡s ♦❢ G ′ ♥❡❡❞❡❞ t♦ ❝♦✈❡r t❤❡ ❡❞❣❡s ✐♥ H ′ ✳ ✽ ✴ ✶✻

Pr♦♦❢ ♦❢ ◆P✲❈♦♠♣❧❡t❡♥❡ss ♦❢ bp ❢♦r ❣❡♥❡r❛❧ ❣r❛♣❤s ❙✉♣♣♦s❡ ✇❡ ❛r❡ ❣✐✈❡♥ ❛ ❣r❛♣❤ G ❛♥❞ ♥❡❡❞ t♦ ✜♥❞ ❛ ✈❡rt❡① ❝♦✈❡r ♦❢ s✐③❡ k � | V ( G ) | ❚r❛♥s❢♦r♠ G ✐♥t♦ G ′ ❜② r❡♣❧❛❝✐♥❣ ❡✈❡r② ❡❞❣❡ ✇✐t❤ ❛ ♣❛t❤ ♦❢ 3 ❡❞❣❡s G ′ ❝♦♥t❛✐♥s ♥♦ 4 ✲❝②❝❧❡s✱ s♦ ♦♥❧② st❛rs ❛r❡ ❜✐❝❧✐q✉❡s ✐♥ G ′ ❚❤✐s ✐♠♣❧✐❡s t❤❛t bp ( G ′ ) = α ( G ′ ) ✱ ✇❤❡r❡ α ( G ′ ) ✐s t❤❡ s✐③❡ ♦❢ t❤❡ ♠✐♥✐♠❛❧ ✈❡rt❡① ❝♦✈❡r ♦❢ G ′ ◆♦t✐❝❡ t❤❛t α ( G ′ ) = α ( G ) + | E | ❚❤✉s✱ bp ( G ′ ) = α ( G ′ ) = α ( G ) + | E | ❙♦ α ( G ) � k ✐❢ ❛♥❞ ♦♥❧② ✐❢ bp ( G ′ ) � k + | E | ✾ ✴ ✶✻