http://www.ebook.edu.vn Chu . o . ng 2 C´ac sˆo ´ co . ba ˙’ n cu ˙’ a d¯ˆo ` thi . 2.1 Chu sˆo ´ Kh´ai niˆe . m m`a ch´ung ta s˜e d¯ˆe ` cˆa . p o . ˙’ d¯ˆay khˆong phu . thuˆo . c v`ao su . . d¯i . nh hu . ´o . ng: ta s˜e n´oi vˆe ` ca . nh ch´u . khˆong pha ˙’ i cung. D - ˆe ˙’ tˆo ˙’ ng qu´at x´et d¯a d¯ˆo ` thi . vˆo hu . ´o . ng G := (V, E) c´o n d¯ı ˙’ nh, m ca . nh v`a p th`anh phˆa ` n liˆen thˆong. D - ˇa . t ρ(G) := n − p, ν(G) := m − ρ(G) = m − n + p. Ta go . i ν(G) l`a chu sˆo ´ cu ˙’ a d¯ˆo ` thi . G. D - i . nh l´y 2.1.1 Cho d¯a d¯ˆo ` thi . vˆo hu . ´o . ng G = (V, E). Gia ˙’ su . ˙’ G l`a d¯ˆo ` thi . nhˆa . n d¯u . o . . c t`u . G bˇa ` ng c´ach nˆo ´ i hai d¯ı ˙’ nh a v`a b cu ˙’ a G bo . ˙’ i mˆo . t ca . nh m´o . i; nˆe ´ u a v`a b tr`ung nhau hoˇa . c c´o thˆe ˙’ nˆo ´ i v´o . i nhau bo . ˙’ i mˆo . t dˆay chuyˆe ` n cu ˙’ a G th`ı ρ(G ) = ρ(G), ν(G ) = ν(G) + 1; trong tru . `o . ng ho . . p ngu . o . . c la . i ρ(G ) = ρ(G) + 1, ν(G ) = ν(G). Ch´u . ng minh. Theo c´ach xˆay du . . ng, d¯a d¯ˆo ` thi . G c´o n = n d¯ı ˙’ nh, m = m + 1 ca . nh v`a gia ˙’ su . ˙’ G c´o p th`anh phˆa ` n liˆen thˆong. Nˆe ´ u a ≡ b hoˇa . c c´o mˆo . t dˆay chuyˆe ` n nˆo ´ i a v´o . i b. Khi d¯´o ph´ep biˆe ´ n d¯ˆo ˙’ i G th`anh G khˆong thay d¯ˆo ˙’ i sˆo ´ th`anh phˆa ` n liˆen thˆong, t´u . c l`a p = p . Do d¯´o ρ(G ) = n − p = n − p = ρ(G), ν(G ) = m − ρ(G ) = ν(G) + 1. 49 http://www.ebook.edu.vn Ngu . o . . c la . i, nˆe ´ u a = b v`a khˆong tˆo ` n ta . i dˆay chuyˆe ` n nˆo ´ i a v`a b, th`ı do c´ach x´ac d¯i . nh G ta c´o p = p − 1. Suy ra ρ(G ) = n − p = n − (p − 1) = n − p + 1 = ρ(G) + 1, ν(G ) = m − ρ(G ) = (m + 1) − (ρ (G) + 1) = m − ρ(G) = ν(G). Hˆe . qua ˙’ 2.1.2 ρ(G) ≥ 0 v`a ν(G) ≥ 0. Ch´u . ng minh. Thˆa . t vˆa . y, xuˆa ´ t ph´at t`u . d¯ˆo ` thi . th`anh lˆa . p bˇa ` ng c´ac d¯ı ˙’ nh cu ˙’ a d¯a d¯ˆo ` thi . vˆo hu . ´o . ng G, d¯ı ˙’ nh no . cˆo lˆa . p v´o . i d¯ı ˙’ nh kia, ta xˆay du . . ng G dˆa ` n dˆa ` n t`u . ng ca . nh mˆo . t; kho . ˙’ i d¯ˆa ` u ta c´o ρ = 0, ν = 0; mˆo ˜ i khi thˆem mˆo . t ca . nh, th`ı hoˇa . c ρ tˇang v`a l´uc d¯´o ν khˆong d¯ˆo ˙’ i, hoˇa . c ν tˇang v`a l´uc d¯´o ρ khˆong d¯ˆo ˙’ i. Nhu . vˆa . y, trong qu´a tr`ınh xˆay du . . ng d¯ˆo ` thi . G , c´ac sˆo ´ ρ v`a ν chı ˙’ c´o thˆe ˙’ tˇang. D - ˆe ˙’ c´o thˆe ˙’ vˆa . n du . ng nh˜u . ng kˆe ´ t qua ˙’ phong ph´u cu ˙’ a d¯a . i sˆo ´ vector trong viˆe . c nghiˆen c´u . u, ngu . `o . i ta thu . `o . ng d¯ˇa . t tu . o . ng ´u . ng mˆo ˜ i chu tr`ınh trong G v´o . i mˆo . t vector theo c´ach sau d¯ˆay. Mˆo ˜ i ca . nh cu ˙’ a d¯a d¯ˆo ` thi . G d¯ˆe ` u d¯u . o . . c d¯i . nh hu . ´o . ng mˆo . t c´ach t`uy ´y; nˆe ´ u chu tr`ınh µ d¯i qua ca . nh e k , r k lˆa ` n thuˆa . n hu . ´o . ng v`a s k lˆa ` n ngu . o . . c hu . ´o . ng th`ı ta d¯ˇa . t c k := r k − s k (nˆe ´ u e k l`a mˆo . t khuyˆen th`ı ta luˆon qui u . ´o . c s k = 0). Vector m chiˆe ` u (c 1 , c 2 , . . . , c m ) go . i l`a vector chu tr`ınh tu . o . ng ´u . ng v´o . i µ v`a k´y hiˆe . u l`a µ (hay l`a µ nˆe ´ u khˆong thˆe ˙’ gˆay ra nhˆa ` m lˆa ˜ n). C´ac chu tr`ınh µ, µ , µ , . . . go . i l`a d¯ˆo . c lˆa . p nˆe ´ u c´ac vector chu tr`ınh tu . o . ng ´u . ng d¯ˆo . c lˆa . p tuyˆe ´ n t´ınh. Ch´u ´y rˇa ` ng, d¯i . nh ngh˜ıa n`ay khˆong phu . thuˆo . c v`ao hu . ´o . ng g´an cho c´ac ca . nh. D - i . nh l´y 2.1.3 Chu sˆo ´ ν(G) cu ˙’ a G = (V, E) bˇa ` ng sˆo ´ cu . . c d¯a . i c´ac chu tr`ınh d¯ˆo . c lˆa . p. Ch´u . ng minh. Tiˆe ´ n h`anh nhu . trong Hˆe . qua ˙’ 2.1.2: d¯ˆa ` u tiˆen ta lˆa ´ y d¯ˆo ` thi . vˆo hu . ´o . ng khˆong c´o ca . nh v´o . i tˆa . p c´ac d¯ı ˙’ nh l`a V. Sau d¯´o ta xˆay du . . ng d¯a d¯ˆo ` thi . G bˇa ` ng c´ach thˆem t`u . ng ca . nh mˆo . t v`ao. Theo D - i . nh l´y 2.1.1, chu sˆo ´ s˜e tˇang mˆo . t d¯o . n vi . nˆe ´ u ca . nh thˆem v`ao lˆa . p ra c´ac chu tr`ınh m´o . i, chu sˆo ´ khˆong thay d¯ˆo ˙’ i trong tru . `o . ng ho . . p ngu . o . . c la . i. Gia ˙’ su . ˙’ , tru . ´o . c khi thˆem ca . nh e k ta d¯˜a c´o mˆo . t co . so . ˙’ gˆo ` m c´ac chu tr`ınh d¯ˆo . c lˆa . p: µ 1 , µ 2 , µ 3 , . . . ; v`a sau khi thˆem ca . nh e k xuˆa ´ t hiˆe . n thˆem c´ac chu tr`ınh so . cˆa ´ p m´o . i γ 1 , γ 2 , . . . , n`ao d¯´o. Hiˆe ˙’ n nhiˆen γ 1 khˆong thˆe ˙’ biˆe ˙’ u diˆe ˜ n tuyˆe ´ n t´ınh qua hˆe . c´ac chu tr`ınh µ j (v`ı c´ac vector 50 http://www.ebook.edu.vn tu . o . ng ´u . ng c´ac chu tr`ınh µ j c´o th`anh phˆa ` n th´u . k bˇa ` ng khˆong, trong khi vector tu . o . ng ´u . ng chu tr`ınh γ 1 c´o th`anh phˆa ` n th´u . k kh´ac khˆong). Mˇa . t kh´ac c´ac vector γ 2 , γ 3 , . . . c´o thˆe ˙’ biˆe ˙’ u diˆe ˜ n tuyˆe ´ n t´ınh qua γ 1 , µ 1 , µ 2 , µ 3 , . . . . T´om la . i mˆo ˜ i khi chu sˆo ´ tˇang mˆo . t d¯o . n vi . th`ı sˆo ´ cu . . c d¯a . i c´ac chu tr`ınh d¯ˆo . c lˆa . p tuyˆe ´ n t´ınh c˜ung tˇang lˆen mˆo . t d¯o . n vi . . D - i . nh l´y d¯u . o . . c ch´u . ng minh. T`u . kˆe ´ t qua ˙’ n`ay, dˆe ˜ d`ang suy ra: Hˆe . qua ˙’ 2.1.4 (a) D - a d¯ˆo ` thi . vˆo hu . ´o . ng G khˆong c´o chu tr`ınh nˆe ´ u v`a chı ˙’ nˆe ´ u ν(G) = 0. (b) D - a d¯ˆo ` thi . vˆo hu . ´o . ng G c´o d¯´ung mˆo . t chu tr`ınh nˆe ´ u v`a chı ˙’ nˆe ´ u ν(G) = 1. D - i . nh l´y 2.1.5 Trong d¯ˆo ` thi . c´o hu . ´o . ng liˆen thˆong ma . nh, chu sˆo ´ bˇa ` ng sˆo ´ cu . . c d¯a . i c´ac ma . ch d¯ˆo . c lˆa . p tuyˆe ´ n t´ınh. Ch´u . ng minh. Thˆa . t vˆa . y, x´et d¯ˆo ` thi . vˆo hu . ´o . ng lˆa . p bo . ˙’ i c´ac cung kh´ac nhau cu ˙’ a G (mˆo ˜ i cung tu . o . ng ´u . ng mˆo . t cˇa . p ca . nh) v`a mˆo . t chu tr`ınh so . cˆa ´ p µ; ta phˆan hoa . ch tˆa . p c´ac d¯ı ˙’ nh trˆen chu tr`ınh n`ay th`anh: tˆa . p S c´ac d¯ı ˙’ nh c´o mˆo . t cung t´o . i n´o v`a mˆo . t cung ra kho ˙’ i n´o, tˆa . p S c´ac d¯ı ˙’ nh c´o hai cung cu ˙’ a µ ra kho ˙’ i n´o v`a tˆa . p S c´ac d¯ı ˙’ nh c´o hai cung cu ˙’ a µ d¯i t´o . i n´o. V`ı sˆo ´ c´ac cung d¯i ra bˇa ` ng sˆo ´ c´ac cung d¯i t´o . i nˆen #S = #S ; gia ˙’ su . ˙’ v 1 , v 2 , . . . , v k l`a c´ac phˆa ` n tu . ˙’ cu ˙’ a S v`a v 1 , v 2 , . . . , v k l`a c´ac phˆa ` n tu . ˙’ cu ˙’ a S . Trˆen chu tr`ınh µ, c´ac phˆa ` n tu . ˙’ cu ˙’ a S v`a cu ˙’ a S xen k˜e nhau v`a ta gia ˙’ su . ˙’ rˇa ` ng sau d¯ı ˙’ nh v i th`ı d¯ı ˙’ nh d¯ˆa ` u tiˆen bˇa ´ t gˇa . p (khˆong thuˆo . c S) l`a v i ; cuˆo ´ i c`ung, nˆe ´ u µ 0 l`a mˆo . t d¯u . `o . ng d¯i gˇa . p d¯ı ˙’ nh x tru . ´o . c d¯ı ˙’ nh y th`ı ta k´y hiˆe . u µ 0 [x, y] l`a d¯u . `o . ng d¯i bˆo . phˆa . n cu ˙’ a µ 0 t`u . x d¯ˆe ´ n y. V`ı d¯ˆo ` thi . liˆen thˆong ma . nh nˆen tˆo ` n ta . i ma . ch µ 1 d¯i qua v i+1 v`a v i v`a d`ung c´ac cung cu ˙’ a µ d¯ˆe ˙’ d¯i t`u . v i+1 d¯ˆe ´ n v i . Chu tr`ınh µ l`a mˆo . t tˆo ˙’ ho . . p tuyˆe ´ n t´ınh cu ˙’ a c´ac ma . ch v`ı ta c´o thˆe ˙’ viˆe ´ t µ = µ[v 1 , v 1 ] − µ 1 [v 2 , v 1 ] + µ[v 2 , v 2 ] + · · · = µ[v 1 , v 1 ] + µ 1 [v 1 , v 2 ] + µ[v 2 , v 2 ] + µ 2 [v 2 , v 3 ] + · · · − (µ 1 + µ 2 + · · · ). Vˆa . y mo . i chu tr`ınh so . cˆa ´ p d¯ˆe ` u l`a tˆo ˙’ ho . . p tuyˆe ´ n t´ınh cu ˙’ a c´ac ma . ch, d¯ˆo ´ i v´o . i c´ac chu tr`ınh bˆa ´ t k`y d¯iˆe ` u d¯´o c˜ung d¯´ung (v`ı n´o l`a tˆo ˙’ ho . . p tuyˆe ´ n t´ınh cu ˙’ a c´ac chu tr`ınh so . cˆa ´ p). Trong R m , c´ac ma . ch lˆa . p th`anh mˆo . t co . so . ˙’ cu ˙’ a khˆong gian vector con sinh bo . ˙’ i c´ac chu tr`ınh, v`a theo D - i . nh l´y 2.1.3 th`ı co . so . ˙’ n`ay c´o sˆo ´ chiˆe ` u l`a ν(G). Vˆa . y sˆo ´ cu . . c d¯a . i c´ac ma . ch d¯ˆo . c lˆa . p tuyˆe ´ n t´ınh bˇa ` ng ν(G). 51 http://www.ebook.edu.vn 2.2 Sˇa ´ c sˆo ´ Gia ˙’ su . ˙’ rˇa ` ng ch´ung ta c´o mˆo . t d¯ˆo ` thi . vˆo hu . ´o . ng G v´o . i n d¯ı ˙’ nh, v`a cˆa ` n tˆo m`au c´ac d¯ı ˙’ nh sao cho hai d¯ı ˙’ nh kˆe ` nhau c´o m`au kh´ac nhau. Hiˆe ˙’ n nhiˆen l`a c´o thˆe ˙’ d`ung n m`au d¯ˆe ˙’ tˆo c´ac d¯ı ˙’ nh d¯´o, nhu . ng nhu . thˆe ´ vˆa ´ n d¯ˆe ` d¯ˇa . t ra la . i khˆong mang t´ınh thu . . c tiˆe ˜ n. Thˆe ´ th`ı sˆo ´ m`au tˆo ´ i thiˆe ˙’ u d¯`oi ho ˙’ i l`a bao nhiˆeu? D - ˆay ch´ınh l`a b`ai to´an tˆo m`au. Khi c´ac d¯ı ˙’ nh d¯u . o . . c tˆo, ch´ung ta c´o thˆe ˙’ nh´om ch´ung v`ao c´ac tˆa . p kh´ac nhau-mˆo . t tˆa . p gˆo ` m c´ac d¯ı ˙’ nh d¯u . o . . c tˆo m`au d¯o ˙’ , mˆo . t tˆa . p c´ac d¯ı ˙’ nh d¯u . o . . c tˆo m`au xanh, vˆan vˆan. D - ˆay ch´ınh l`a b`ai to´an phˆan hoa . ch. B`ai to´an tˆo m`au v`a phˆan hoa . ch d˜ı nhiˆen c´o thˆe ˙’ x´et trˆen c´ac ca . nh cu ˙’ a d¯ˆo ` thi . . Trong tru . `o . ng ho . . p d¯ˆo ` thi . phˇa ˙’ ng thˆa . m ch´ı c´o thˆe ˙’ quan tˆam d¯ˆe ´ n viˆe . c tˆo m`au c´ac diˆe . n. Trong phˆa ` n n`ay ta chı ˙’ x´et c´ac d¯ˆo ` thi . vˆo hu . ´o . ng liˆen thˆong. D - i . nh ngh˜ıa 2.2.1 Cho tru . ´o . c mˆo . t sˆo ´ nguyˆen p, ta n´oi rˇa ` ng d¯ˆo ` thi . G l`a p−sˇa ´ c nˆe ´ u bˇa ` ng p m`au kh´ac nhau ta c´o thˆe ˙’ tˆo m`au c´ac d¯ı ˙’ nh, sao cho hai d¯ı ˙’ nh kˆe ` nhau khˆong c`ung mˆo . t m`au. Sˆo ´ p nho ˙’ nhˆa ´ t, m`a d¯ˆo ´ i v´o . i sˆo ´ d¯´o G l`a p−sˇa ´ c go . i l`a sˇa ´ c sˆo ´ cu ˙’ a d¯ˆo ` thi . G v`a k´y hiˆe . u l`a γ(G). V´ı du . 2.2.2 H`ınh 2.1 minh ho . a ba c´ach tˆo m`au kh´ac nhau cu ˙’ a d¯ˆo ` thi . . Dˆe ˜ d`ang kiˆe ˙’ m tra rˇa ` ng d¯ˆo ` thi . n`ay l`a 2−sˇa ´ c. b r • • . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . b r • • . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . b r • • . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . g y p • • • (a) g y r • • • (b) r r b • • • (c) H`ınh 2.1: V´ı du . 2.2.3 Ba ˙’ n d¯ˆo ` d¯i . a l´y. Ta v˜e trˆen mˇa . t phˇa ˙’ ng mˆo . t ba ˙’ n d¯ˆo ` . Go . i V l`a tˆa . p ho . . p c´ac nu . ´o . c, d¯ˇa . t (i, j) ∈ E nˆe ´ u c´ac nu . ´o . c i v`a j c´o biˆen gi´o . i chung. D - ˆo ` thi . G = (V, E) d¯ˆo ´ i x ´u . ng v`a c´o t´ınh chˆa ´ t rˆa ´ t d¯ˇa . c biˆe . t l`a: c´o thˆe ˙’ v˜e n´o lˆen mˇa . t phˇa ˙’ ng m`a khˆong c´o hai ca . nh n`ao cˇa ´ t nhau (tr`u . ta . i c´ac d¯ı ˙’ nh chung); n´oi c´ach kh´ac, G l`a d¯ˆo ` thi . phˇa ˙’ ng. Ngu . `o . i ta d¯˜a biˆe ´ t rˇa ` ng sˇa ´ c sˆo ´ cu ˙’ a mo . i d¯ˆo ` thi . phˇa ˙’ ng d¯ˆe ` u nho ˙’ ho . n hoˇa . c bˇa ` ng bˆo ´ n (D - i . nh l´y 6.4.7). Nhu . vˆa . y bˇa ` ng bˆo ´ n m`au c˜ung d¯u ˙’ d¯ˆe ˙’ tˆo m`au ba ˙’ n d¯ˆo ` phˇa ˙’ ng sao cho hai nu . ´o . c kˆe ` nhau khˆong c`ung mˆo . t m`au. 52 http://www.ebook.edu.vn T`u . d¯i . nh ngh˜ıa dˆe ˜ d`ang suy ra 1. Mˆo . t d¯ˆo ` thi . chı ˙’ c´o c´ac d¯ı ˙’ nh cˆo lˆa . p l`a 1−sˇa ´ c. 2. Mˆo . t d¯ˆo ` thi . c´o mˆo . t hoˇa . c hai ca . nh (khˆong pha ˙’ i l`a mˆo . t khuyˆen) c´o sˇa ´ c sˆo ´ ´ıt nhˆa ´ t bˇa ` ng hai. 3. D - ˆo ` thi . d¯ˆa ` y d¯u ˙’ n d¯ı ˙’ nh K n l`a n−sˇa ´ c. 4. D - ˆo ` thi . l`a mˆo . t chu tr`ınh d¯o . n gia ˙’ n v´o . i n d¯ı ˙’ nh, n > 3, l`a 2−sˇa ´ c nˆe ´ u n chˇa ˜ n v`a 3−sˇa ´ c nˆe ´ u n le ˙’ . 5. Hiˆe ˙’ n nhiˆen, mo . i d¯ˆo ` thi . 2−sˇa ´ c l`a hai phˆa ` n do ch´ung ta c´o thˆe ˙’ phˆan hoa . ch tˆa . p c´ac d¯ı ˙’ nh V th`anh hai tˆa . p con theo m`au d¯u . o . . c tˆo trˆen c´ac d¯ı ˙’ nh. Tu . o . ng tu . . , d¯ˆo ` thi . hai phˆa ` n l`a 2−sˇa ´ c, v´o . i mˆo . t tru . `o . ng ho . . p ngoa . i lˆe . tˆa ` m thu . `o . ng: d¯ˆo ` thi . c´o ´ıt nhˆa ´ t hai d¯ı ˙’ nh cˆo lˆa . p v`a khˆong c´o ca . nh l`a hai phˆa ` n nhu . ng l`a 1−sˇa ´ c. D - i . nh ngh˜ıa 2.2.4 Ta go . i sˇa ´ c l´o . p cu ˙’ a d¯ˆo ` thi . G l`a sˆo ´ nguyˆen q c´o c´ac t´ınh chˆa ´ t sau: 1. C´o thˆe ˙’ d`ung q m`au kh´ac nhau d¯ˆe ˙’ tˆo m`au c´ac ca . nh cu ˙’ a G sao cho hai ca . nh kˆe ` nhau khˆong c`ung mˆo . t m`au; 2. D - iˆe ` u n`ay khˆong thˆe ˙’ l`am d¯u . o . . c v´o . i (q − 1) m`au. Nhˆa . n x´et rˇa ` ng sˇa ´ c l´o . p cu ˙’ a d¯ˆo ` thi . G = (V, E) ch´ınh l`a sˇa ´ c sˆo ´ cu ˙’ a d¯ˆo ` thi . G = (V , E ) d¯u . o . . c x´ac d¯i . nh nhu . sau: mˆo ˜ i d¯ı ˙’ nh cu ˙’ a G tu . o . ng ´u . ng mˆo . t ca . nh cu ˙’ a G; ca . nh e = (v 1 , v 2 ) ∈ E nˆe ´ u c´ac ca . nh e 1 v`a e 2 (tu . o . ng ´u . ng v´o . i hai d¯ı ˙’ nh v 1 , v 2 ) kˆe ` nhau. Nhu . vˆa . y b`ai to´an sˇa ´ c l´o . p d¯u . a vˆe ` b`ai to´an sˇa ´ c sˆo ´ . Du . ´o . i d¯ˆay l`a mˆo . t v`ai kˆe ´ t qua ˙’ co . ba ˙’ n vˆe ` sˇa ´ c sˆo ´ . D - i . nh l´y 2.2.5 [K¨onig] D - ˆo ` thi . vˆo hu . ´o . ng G l`a 2−sˇa ´ c nˆe ´ u v`a chı ˙’ nˆe ´ u n´o khˆong c´o chu tr`ınh c´o d¯ˆo . d`ai le ˙’ . Ch´u . ng minh D - iˆe ` u kiˆe . n cˆa ` n. Nˆe ´ u d¯ˆo ` thi . G l`a 2−sˇa ´ c, th`ı tˆa ´ t nhiˆen G khˆong ch´u . a chu tr`ınh c´o d¯ˆo . d`ai le ˙’ , v`ı c´ac d¯ı ˙’ nh cu ˙’ a mˆo . t chu tr`ınh loa . i nhu . vˆa . y khˆong thˆe ˙’ tˆo bˇa ` ng hai m`au theo nhu . quy tˇa ´ c d¯˜a chı ˙’ ra o . ˙’ trˆen. D - iˆe ` u kiˆe . n d¯u ˙’ . Gia ˙’ su . ˙’ d¯ˆo ` thi . G khˆong c´o chu tr`ınh c´o d¯ˆo . d`ai le ˙’ , ta ch´u . ng minh n´o l`a 2−sˇa ´ c. Khˆong gia ˙’ m tˆo ˙’ ng qu´at coi G l`a liˆen thˆong. Ta s˜e tˆo m`au dˆa ` n c´ac d¯ı ˙’ nh cu ˙’ a G theo quy tˇa ´ c sau: 53 http://www.ebook.edu.vn • tˆo m`au xanh cho mˆo . t d¯ı ˙’ nh a n`ao d¯´o; • nˆe ´ u mˆo . t d¯ı ˙’ nh x n`ao d¯´o d¯˜a d¯u . o . . c tˆo xanh, th`ı ta tˆo d¯o ˙’ tˆa ´ t ca ˙’ c´ac d¯ı ˙’ nh kˆe ` v´o . i n´o; nˆe ´ u d¯ı ˙’ nh y d¯˜a d¯u . o . . c tˆo d¯o ˙’ , th`ı ta tˆo xanh tˆa ´ t ca ˙’ c´ac d¯ı ˙’ nh kˆe ` v´o . i y. V`ı d¯ˆo ` thi . G liˆen thˆong, nˆen s´o . m hay muˆo . n th`ı mo . i d¯ı ˙’ nh cu ˙’ a n´o d¯ˆe ` u d¯u . o . . c tˆo m`au hˆe ´ t, v`a mˆo . t d¯ı ˙’ nh x khˆong thˆe ˙’ c`ung mˆo . t l´uc d¯u . o . . c tˆo xanh v`a tˆo d¯o ˙’ , v`ı nhu . vˆa . y th`ı x v`a a s˜e c`ung nˇa ` m trˆen mˆo . t chu tr`ınh c´o d¯ˆo . d`ai le ˙’ . Vˆa . y d¯ˆo ` thi . G l`a 2−sˇa ´ c. Ch´u ´y rˇa ` ng t´ınh chˆa ´ t d¯ˆo ` thi . G khˆong c´o chu tr`ınh v´o . i d¯ˆo . d`ai le ˙’ tu . o . ng d¯u . o . ng v´o . i t´ınh chˆa ´ t G khˆong c´o chu tr`ınh so . cˆa ´ p v´o . i d¯ˆo . d`ai le ˙’ . Thˆa . t vˆa . y gia ˙’ su . ˙’ c´o mˆo . t chu tr`ınh µ = {v 0 , v 1 , . . . , v p = v 0 } c´o d¯ˆo . d`ai p le ˙’ . Mˆo ˜ i khi gˇa . p hai d¯ı ˙’ nh v j v`a v k v´o . i j < k < p v`a v j = v k , ta phˆan chia µ th`anh hai chu tr`ınh bˆo . phˆa . n µ 1 = {v j , . . . , v k } v`a µ 2 = {v 0 , . . . , v j , v k , . . . , v 0 }; ho . n n˜u . a mˆo . t trong hai chu tr`ınh c´o d¯ˆo . d`ai le ˙’ (v`ı nˆe ´ u khˆong nhu . thˆe ´ th`ı µ s˜e c´o d¯ˆo . d`ai chˇa ˜ n). Ta thˆa ´ y rˇa ` ng nˆe ´ u tiˆe ´ p tu . c phˆan chia chu tr`ınh µ theo c´ach d¯´o cho d¯ˆe ´ n khi c`on c´o thˆe ˙’ l`am d¯u . o . . c, th`ı mˆo ˜ i lˆa ` n vˆa ˜ n c`on d¯u . o . . c mˆo . t chu tr`ınh c´o d¯ˆo . d`ai le ˙’ ; v`ı cuˆo ´ i c`ung mo . i chu tr`ınh d¯ˆe ` u l`a so . cˆa ´ p, nˆen xa ˙’ y ra mˆau thuˆa ˜ n; v`a ta c´o d¯iˆe ` u cˆa ` n ch´u . ng minh. D - i . nh l´y sau d¯ˆay cho ta biˆe ´ t cˆa . n trˆen cu ˙’ a sˇa ´ c sˆo ´ . D - i . nh l´y 2.2.6 K´y hiˆe . u d max l`a bˆa . c cu . . c d¯a . i cu ˙’ a c´ac d¯ı ˙’ nh trong G. Khi d¯´o γ(G) ≤ 1 + d max . Ch´u . ng minh. B`ai tˆa . p. Brooks [9] d¯˜a ch´u . ng minh rˇa ` ng nˆe ´ u G l`a d¯ˆo ` thi . khˆong d¯ˆa ` y d¯u ˙’ , c´o d max d¯ı ˙’ nh th`ı γ(G) ≤ d max . 2.2.1 C´ach t`ım sˇa ´ c sˆo ´ X´et d¯ˆo ` thi . G = (V, Γ) c´o n d¯ı ˙’ nh v`a m ca . nh; muˆo ´ n t`ım sˇa ´ c sˆo ´ cu ˙’ a n´o ta c´o thˆe ˙’ d`ung mˆo . t phu . o . ng ph´ap thu . . c nghiˆe . m rˆa ´ t d¯o . n gia ˙’ n, ´ap du . ng tru . . c tiˆe ´ p d¯u . o . . c, nhu . ng khˆong pha ˙’ i l ´uc n`ao c˜ung c´o hiˆe . u qua ˙’ hoˇa . c c´o thˆe ˙’ d`ung phu . o . ng ph´ap gia ˙’ i t´ıch, n´o cho ta mˆo . t l`o . i gia ˙’ i hˆe . thˆo ´ ng, nhu . ng n´oi chung cˆa ` n m´ay t´ınh d¯iˆe . n tu . ˙’ . 54 http://www.ebook.edu.vn Phu . o . ng ph´ap thu . . c nghiˆe . m Bˇa ` ng c´ach tˆo m`au t`uy ´y d`ung c´ac m`au 1, 2, . . . , p v`a t`ım c´ach loa . i dˆa ` n mˆo . t m`au n`ao d¯´o (go . i l`a “m`au t´o . i ha . n”) trong c´ac m`au ˆa ´ y. Muˆo ´ n vˆa . y, ta x´et d¯ı ˙’ nh v c´o m`au t´o . i ha . n d¯´o v`a c´ac th`anh phˆa ` n liˆen thˆong C jk 1 , C jk 2 , . . . cu ˙’ a d¯ˆo ` thi . con sinh ra bo . ˙’ i hai m`au khˆong t´o . i ha . n j v`a k. Ta c´o thˆe ˙’ lˆa . p t´u . c thay m`au cu ˙’ a d¯ı ˙’ nh v nˆe ´ u c´ac tˆa . p ho . . p C jk 1 ∩ Γ(v), C jk 2 ∩ Γ(v), . . . khˆong pha ˙’ i l`a hai m`au: lˆa ´ y riˆeng r˜e mˆo ˜ i th`anh phˆa ` n C jk m`a v´o . i th`anh phˆa ` n d¯´o th`ı c´ac d¯ı ˙’ nh cu ˙’ a C jk ∩ Γ(v) c´o m`au j, ho´an vi . trong c´ac th`anh phˆa ` n d¯´o c´ac m`au j v`a k (khˆong thay d¯ˆo ˙’ i m`au cu ˙’ a c´ac d¯ı ˙’ nh kh´ac), cuˆo ´ i c`ung tˆo d¯ı ˙’ nh v m`au j (l´uc n`ay d¯ı ˙’ nh v khˆong kˆe ` v´o . i d¯ı ˙’ nh n`ao c´o m`au j). Phu . o . ng ph´ap gia ˙’ i t´ıch Kiˆe ˙’ m tra bˇa ` ng c´ach gia ˙’ i t´ıch xem d¯ˆo ` thi . G c´o thˆe ˙’ d¯u . o . . c tˆo bˇa ` ng p m`au d¯u . o . . c khˆong. Phu . o . ng ph´ap d¯´o nhu . sau: V´o . i mˆo ˜ i c´ach tˆo bˇa ` ng p m`au, ta cho tu . o . ng ´u . ng v´o . i mˆo . t hˆe . thˆo ´ ng c´ac sˆo ´ x ij , i = 1, 2, . . . , n; j = 1, 2, . . . , p, d¯u . o . . c x´ac d¯i . nh nhu . sau: x ij = 1 nˆe ´ u d¯ı ˙’ nh i c´o m`au j, 0 trong tru . `o . ng ho . . p ngu . o . . c la . i. D - ˇa . t r ij = 1 nˆe ´ u ca . nh e j liˆen thuˆo . c d¯ı ˙’ nh v i , 0 trong tru . `o . ng ho . . p ngu . o . . c la . i. L´uc d¯´o b`ai to´an tro . ˙’ th`anh t`ım c´ac sˆo ´ nguyˆen x ij sao cho: x ij ≥ 0, p q=1 x iq ≤ 1, i = 1, 2, . . . , n, n k=1 r jk x kq ≤ 1, j = 1, 2, . . . , m; q = 1, 2, . . . , p. Ta c´o hˆe . bˆa ´ t d¯ˇa ˙’ ng th´u . c tuyˆe ´ n t´ınh. Dˆe ˜ r`ang thˆa ´ y rˇa ` ng hˆe . d¯´o th´ıch ho . . p v´o . i c´ac phu . o . ng ph´ap thˆong thu . `o . ng cu ˙’ a quy hoa . ch nguyˆen v`a do d¯´o c´o thˆe ˙’ d`ung phu . o . ng ph´ap cu ˙’ a Gomory (du . . a trˆen phu . o . ng ph´ap d¯o . n h`ınh cu ˙’ a Dantzig) d¯ˆe ˙’ gia ˙’ i. 2.3 Sˆo ´ ˆo ˙’ n d¯i . nh trong V´o . i d¯ˆo ` thi . G = (V, Γ) cho tru . ´o . c ta thu . `o . ng quan tˆam d¯ˆe ´ n tˆa . p con cu ˙’ a V c´o nh˜u . ng t´ınh chˆa ´ t n`ao d¯´o. Chˇa ˙’ ng ha . n, t`ım mˆo . t tˆa . p con S ⊂ V c´o sˆo ´ phˆa ` n tu . ˙’ l´o . n nhˆa ´ t sao cho d¯ˆo ` thi . con sinh 55 http://www.ebook.edu.vn bo . ˙’ i S l`a d¯ˆa ` y d¯u ˙’ ? Hoˇa . c t`ım tˆa . p con c´ac d¯ı ˙’ nh cu ˙’ a d¯ˆo ` thi . G c´o sˆo ´ phˆa ` n tu . ˙’ nhiˆe ` u nhˆa ´ t sao cho hai d¯ı ˙’ nh trong d¯´o khˆong kˆe ` nhau. Mˆo . t b`ai to´an kh´ac l`a t`ım tˆa . p con S cu ˙’ a V c´o sˆo ´ phˆa ` n tu . ˙’ ´ıt nhˆa ´ t sao cho mo . i d¯ı ˙’ nh thuˆo . c V \ S kˆe ` v´o . i mˆo . t d¯ı ˙’ nh trong S. C´ac sˆo ´ v`a c´ac tˆa . p con tu . o . ng ´u . ng l`o . i gia ˙’ i c´ac b`ai to´an trˆen cho nh˜u . ng t´ınh chˆa ´ t quan trong cu ˙’ a d¯ˆo ` thi . v`a c´o nhiˆe ` u ´u . ng du . ng tru . . c tiˆe ´ p trong b`ai to´an lˆa . p li . ch, phˆan t´ıch cluster, phˆan loa . i sˆo ´ , xu . ˙’ l´y song song trˆen m´ay t´ınh, vi . tr´ı thuˆa . n lo . . i v`a thay thˆe ´ c´ac th`anh phˆa ` n d¯iˆe . n tu . ˙’ , v.v. D - i . nh ngh˜ıa 2.3.1 X´et d¯ˆo ` thi . vˆo hu . ´o . ng G := (V, Γ); tˆa . p ho . . p S ⊂ V d¯u . o . . c go . i l`a tˆa . p ho . . p ˆo ˙’ n d¯i . nh trong nˆe ´ u hai d¯ı ˙’ nh bˆa ´ t k`y cu ˙’ a S d¯ˆe ` u khˆong kˆe ` nhau; n´oi c´ach kh´ac, v´o . i mo . i cˇa . p d¯ı ˙’ nh a, b ∈ S th`ı b /∈ Γ(a). K´y hiˆe . u S l`a ho . c´ac tˆa . p ˆo ˙’ n d¯i . nh trong cu ˙’ a d¯ˆo ` thi . G. Khi d¯´o 1. Tˆa . p trˆo ´ ng ∅ thuˆo . c S. 2. Nˆe ´ u S ∈ S v`a A ⊂ S th`ı A ∈ S. N´oi c´ach kh´ac, tˆa . p con cu ˙’ a mˆo . t tˆa . p ˆo ˙’ n d¯i . nh trong c˜ung l`a mˆo . t tˆa . p ˆo ˙’ n d¯i . nh trong. Tˆa . p ˆo ˙’ n d¯i . nh trong l`a cu . . c d¯a . i nˆe ´ u thˆem mˆo . t d¯ı ˙’ nh bˆa ´ t k`y v`ao n´o th`ı s˜e khˆong c`on ˆo ˙’ n d¯i . nh trong n˜u . a. D - a . i lu . o . . ng α(G) := max{#S | S ∈ S} d¯u . o . . c go . i l`a sˆo ´ ˆo ˙’ n d¯i . nh trong cu ˙’ a G. V´ı du . 2.3.2 X´et d¯ˆo ` thi . trong H`ınh 2.2. Tˆa . p c´ac d¯ı ˙’ nh {v 7 , v 8 , v 2 } l`a ˆo ˙’ n d¯i . nh trong nhu . ng khˆong cu . . c d¯a . i; tˆa . p {v 7 , v 8 , v 2 , v 5 } l`a ˆo ˙’ n d¯i . nh trong cu . . c d¯a . i. C´ac tˆa . p {v 1 , v 3 , v 7 } v`a {v 4 , v 6 } c˜ung l`a tˆa . p ˆo ˙’ n d¯i . nh trong cu . . c d¯a . i v`a do d¯´o, n´oi chung c´o thˆe ˙’ c´o nhiˆe ` u tˆa . p ˆo ˙’ n d¯i . nh trong cu . . c d¯a . i. Ho . c´ac tˆa . p ˆo ˙’ n d¯i . nh trong cu . . c d¯a . i cu ˙’ a d¯ˆo ` thi . n`ay l`a {v 7 , v 8 , v 2 , v 5 }, {v 1 , v 3 , v 7 }, {v 4 , v 6 }, {v 3 , v 6 }, {v 1 , v 5 , v 7 }, {v 1 , v 4 }, {v 3 , v 7 , v 8 }. V´ı du . 2.3.3 [Gauss] B`ai to´an t´am con hˆa . u. Trˆen b`an c`o . c´o thˆe ˙’ bˆo ´ tr´ı t´am con hˆa . u, sao cho khˆong c´o con n`ao ch´em d¯u . o . . c con n`ao khˆong? B`ai to´an nˆo ˙’ i tiˆe ´ ng n`ay d¯u . a vˆe ` t`ım mˆo . t tˆa . p ho . . p ˆo ˙’ n d¯i . nh trong cu . . c d¯a . i cu ˙’ a d¯ˆo ` thi . vˆo hu . ´o . ng c´o 64 d¯ı ˙’ nh (l`a c´ac ˆo trˆen b`an c`o . ), trong d¯´o y ∈ Γ(x) nˆe ´ u c´ac ˆo x v`a y nˇa ` m trˆen c`ung mˆo . t h`ang, mˆo . t cˆo . t hay mˆo . t d¯u . `o . ng ch´eo. Thu . . c tˆe ´ , kh´o khˇan la . i l´o . n ho . n l`a khi ngu . `o . i ta m´o . i thoa . t nh`ın: l´uc d¯ˆa ` u Gauss tu . o . ˙’ ng c´o 76 l`o . i 56 http://www.ebook.edu.vn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v 1 v 2 v 3 v 4 v 5 v 6 v 7 v 8 • • • • • • • • H`ınh 2.2: gia ˙’ i, c`on t`o . b´ao vˆe ` c`o . o . ˙’ Berlin“Schachzeitung” nˇam 1854 chı ˙’ m´o . i d¯u . a ra 40 l`o . i gia ˙’ i thˆe ´ c`o . do c´ac nh`a ham c`o . t`ım ra. Su . . thˆa . t th`ı c´o 92 l`o . i gia ˙’ i nhu . 12 so . d¯ˆo ` du . ´o . i d¯ˆay: (72631485) (61528374) (58417263) (35841726) (46152837) (57263148) (16837425) (57263184) (48157263) (51468273) (42751863) (35281746) Mˆo ˜ i so . d¯ˆo ` trˆen tu . o . ng ´u . ng v´o . i mˆo . t ho´an vi . , v`a t`u . mˆo . t so . d¯ˆo ` ta suy ra t´am l`o . i gia ˙’ i kh´ac nhau: ba l`o . i gia ˙’ i bˇa ` ng c´ach quay 90 0 , 180 0 v`a 270 0 ; c´ac l`o . i gia ˙’ i kh´ac suy ra bˇa ` ng c´ach d¯ˆo ´ i x´u . ng mˆo ˜ i so . d¯ˆo ` nhˆa . n d¯u . o . . c qua d¯u . `o . ng ch´eo ch´ınh; ho´an vi . cuˆo ´ i c`ung (35281746) chı ˙’ c´o bˆo ´ n l`o . i gia ˙’ i v`ı so . d¯ˆo ` tu . o . ng ´u . ng s˜e tr`ung v´o . i ch´ınh n´o sau khi quay 180 0 . V´ı du . 2.3.4 B`e cu ˙’ a mˆo . t d¯ˆo ` thi . G l`a tˆa . p ho . . p C ⊂ V sao cho a ∈ C, b ∈ C suy ra b ∈ Γ(a). Nˆe ´ u V l`a mˆo . t tˆa . p ho . . p ngu . `o . i, v`a b ∈ Γ(a) chı ˙’ su . . d¯ˆo ` ng t`ınh gi˜u . a a v`a b, th`ı thu . `o . ng yˆeu cˆa ` u t`ım b`e cu . . c d¯a . i, ngh˜ıa l`a hˆo . i c´o sˆo ´ ngu . `o . i tham gia nhiˆe ` u nhˆa ´ t. X´et d¯ˆo ` thi . G := (V, Γ ) x´ac d¯i . nh nhu . sau: b ∈ Γ(a ) nˆe ´ u v`a chı ˙’ nˆe ´ u b /∈ Γ(a). B`ai to´an quy vˆe ` t`ım mˆo . t tˆa . p ho . . p ˆo ˙’ n d¯i . nh trong cu . . c d¯a . i cu ˙’ a d¯ˆo ` thi . (V, Γ ). V´ı du . 2.3.5 B`ai to´an vˆe ` c´ac cˆo g´ai l`a d¯ˆo ´ i tu . o . . ng cu ˙’ a nhiˆe ` u cˆong tr`ınh to´an ho . c, c´o thˆe ˙’ ph´at biˆe ˙’ u nhu . sau: mˆo . t k´y t´uc x´a nuˆoi du . ˜o . ng 15 cˆo g´ai, k´y hiˆe . u l`a a, b, c, d, e, f, g, h, i, j, k, l, m, n, o; h`ang ng`ay c´ac cˆo d¯i da . o cho . i th`anh t`u . ng bˆo . ba, c´o thˆe ˙’ d¯u . a c´ac cˆo d¯i cho . i trong ba ˙’ y ng`ay liˆe ` n sao cho khˆong c´o hai cˆo n`ao c`ung d¯i trong mˆo . t bˆo . ba qu´a mˆo . t lˆa ` n d¯u . o . . c khˆong? V´o . i nh˜u . ng nhˆa . n x´et vˆe ` d¯ˆo ´ i x´u . ng, Cayley d¯˜a t`ım ra l`o . i gia ˙’ i nhu . sau: 57 http://www.ebook.edu.vn Chu ˙’ Nhˆa . t Th´u . Hai Th´u . Ba Th´u . Tu . Th´u . Nˇam Th´u . S´au Th´u . Ba ˙’ y afk abe al m ado agn ahj aci bgl cno bcf bik bdj bmn bho chm dfl deh cjl cek cdg dkm din ghk gio egm fmo fei eln ejo ijm jkn fhn hil klo fgj B`ai to´an vˆe ` c´ac cˆo g´ai c`ung loa . i v´o . i mˆo . t b`ai to´an th´u . hai c˜ung nˆo ˙’ i tiˆe ´ ng go . i l`a b`ai to´an bˆo ˙’ tro . . : v´o . i 15 cˆo g´ai c´o thˆe ˙’ lˆa ` n lu . o . . t lˆa . p 35 bˆo . ba kh´ac nhau sao cho khˆong c´o hai cˆo n`ao c`ung trong mˆo . t bˆo . ba qu´a mˆo . t lˆa ` n khˆong? D - ˆe ˙’ gia ˙’ i b`ai to´an bˆo ˙’ tro . . , chı ˙’ cˆa ` n lˆa . p d¯ˆo ` thi . G c´o c´ac d¯ı ˙’ nh l`a 455 bˆo . ba c´o thˆe ˙’ c´o, hai bˆo . ba d¯u . o . . c xem l`a kˆe ` nhau nˆe ´ u ch´ung c´o chung hai cˆo g´ai: khi d¯´o cˆa ` n t`ım mˆo . t tˆa . p ho . . p ˆo ˙’ n d¯i . nh trong cu . . c d¯a . i. Ta c´o α (G) ≤ 35 v`ı mˆo . t cˆo g´ai chı ˙’ c´o mˇa . t nhiˆe ` u nhˆa ´ t l`a trong 7 bˆo . ba kh´ac nhau, nˆen tˆa ´ t ca ˙’ c´o 15 × 7 × 1 3 = 35 bˆo . l`a nhiˆe ` u nhˆa ´ t; v`ı vˆa . y mo . i tˆa . p ho . . p ˆo ˙’ n d¯i . nh trong c´o 35 bˆo . ba d¯ˆe ` u l`a cu . . c d¯a . i. D - ˆe ˙’ x´et xem mˆo . t l`o . i gia ˙’ i n`ao d¯´o cu ˙’ a b`ai to´an bˆo ˙’ tro . . c´o cho l`o . i gia ˙’ i cu ˙’ a b`ai to´an vˆe ` c´ac cˆo g´ai khˆong, ta lˆa . p d¯ˆo ` thi . G c´o c´ac d¯ı ˙’ nh l`a 35 bˆo . ba cho nghiˆe . m cu ˙’ a b`ai to´an bˆo ˙’ tro . . , hai bˆo . ba d¯u . o . . c xem l`a kˆe ` nhau nˆe ´ u c´o chung mˆo . t cˆo g´ai; nˆe ´ u sˇa ´ c sˆo ´ γ(G ) > 7 th`ı pha ˙’ i cho . n tˆa . p ho . . p kh´ac gˆo ` m c´ac bˆo . ba cho nghiˆe . m cu ˙’ a b`ai to´an bˆo ˙’ tro . . . Dˆe ˜ d`ang kiˆe ˙’ m tra rˇa ` ng tˆo ` n ta . i nh˜u . ng l`o . i gia ˙’ i cu ˙’ a b`ai to´an bˆo ˙’ tro . . khˆong dˆa ˜ n d¯ˆe ´ n l`o . i gia ˙’ i cu ˙’ a b`ai to´an vˆe ` c´ac cˆo g´ai. Nhˆa . n x´et 2.3.6 (a) Sˇa ´ c sˆo ´ γ(G) v`a sˆo ´ ˆo ˙’ n d¯i . nh trong α(G) liˆen hˆe . v´o . i nhau bˇa ` ng bˆa ´ t d¯ˇa ˙’ ng th´u . c γ(G) × α(G) ≥ n. Thˆa . t vˆa . y, c´o thˆe ˙’ phˆan hoa . ch V th`anh γ := γ(G) tˆa . p ho . . p ˆo ˙’ n d¯i . nh trong, lˆa . p bo . ˙’ i c´ac d¯ı ˙’ nh c`ung m`au v`a tu . o . ng ´u . ng ch´u . a n 1 , n 2 , . . . , n γ d¯ı ˙’ nh. Vˆa . y ta c´o n = n 1 + n 2 + · · · + n γ ≤ α(G) + α(G) + · · · + α(G) = γ(G).α(G). (b) Ta c´o thˆe ˙’ ho ˙’ i: pha ˙’ i chˇang mˆo ´ i liˆen hˆe . gi˜u . a hai kh´ai niˆe . m l`a chu . a chˇa . t ch˜e. Pha ˙’ i chˇang c´o thˆe ˙’ t`ım sˇa ´ c sˆo ´ bˇa ` ng c´ach: tru . ´o . c hˆe ´ t d`ung m`au (1) d¯ˆe ˙’ tˆo tˆa . p ˆo ˙’ n d¯i . nh trong cu . . c d¯a . i S 1 . Rˆo ` i d`ung m`au (2) d¯ˆe ˙’ tˆo tˆa . p ˆo ˙’ n d¯i . nh trong cu . . c d¯a . i S 2 cu ˙’ a d¯ˆo ` thi . con sinh ra bo . ˙’ i c´ac d¯ı ˙’ nh cu ˙’ a V \ S 1 , sau d´o d`ung m`au (3) d¯ˆe ˙’ tˆo tˆa . p ho . . p ˆo ˙’ n d¯i . nh trong cu . . c d¯a . i cu ˙’ a d¯ˆo ` thi . con c`on la . i, v.v. Thu . . c ra khˆong pha ˙’ i nhu . vˆa . y. D - ˆo ` thi . trˆen H`ınh 2.3 (c´o sˇa ´ c sˆo ´ bˇa ` ng 4) ch´u . ng to ˙’ d¯iˆe ` u 58 [...]... ˙ ¯´ ınh o a u ¯ u ´ o ´ ˙ ˙ ’ V´ du 2. 6 .2 Khˆng phai moi d` thi d` u c´ nhˆn Nˆu d` thi c´ nhˆn S0 th` c´c sˆ ˆ’n ı o ˆ e e ¯ˆ o a ı a o o ¯o ¯ˆ o a ` u kiˆn: ˙ o ’ ˙ a ¯e ’ d nh cua n´ thoa m˜n d iˆ ¯i e α(G) = max #S ≥ #S0 ≥ min = β(G) S∈S T ∈T - D` thi trong H` 2. 14 khˆng c´ nhˆn v` o ınh o o a ı α(G) = 1 < 2 = β(G) Tr´i lai, d` thi trong H` 2. 15 c´ hai nhˆn l` S = {b, d} v` S = {a,... ¯ a o - ˙ e a ˙ e ´ ´ ´ ˙ ’ ˙ ’ Dinh l´ 2. 5.3 Phu g cu a mˆt d` thi l` tˆi thiˆ’u nˆu v` chı’ nˆu g khˆng ch´.a c´c dˆy y o ¯ˆ a o e o u a a o n ho.n hoˇc bˇ ng ba ` o ¯o a o chuyˆn c´ d ˆ d`i l´ e a ` a - ` ˙ ˙ ` o ˙ ˙ ’ ’ ˙ o ’ ´ ’ Ch´.ng minh Diˆu kiˆn cˆn Gia su g l` phu tˆi thiˆ’u cua G m` ch´.a mˆt dˆy chuyˆn c´ u e e ` a e a u o a e a d o d`i ba l` ¯ˆ a a µ := {v1 , e1 , v2 , e2 , v3... v6 • v4 • • v5 • v8 • v3 • v2 • v1 • v9 H` 2. 7: ınh ˙ ` ınh e ´ ’ ˙ o u ’ V´ du 2. 4.5 Vi tr´ cua c´c “tˆm” dˆ’ phu mˆt v`ng C´ nhiˆu l˜ vu.c liˆn quan dˆn b`i ı a ¯e o e ¯e a ı ˙ a to´n n`y: a a ` ¯e ´ ’ 1 Vi tr´ cua c´c m´y ph´t T.V hay radio truyˆn dˆn mˆt sˆ vi tr´ cho tru.´.c a a e o o ı o ı ˙ a ´ 62 http://www.ebook.edu.vn ˙ ’ 2 Vi tr´ cua c´c tram g´c dˆ’ gi´m s´t mˆt v`ng... o e a a o ¯ a - inh l´ 2. 6.4 S l` nhˆn cua d` thi d a cho ˙ ¯o ¯˜ ’ ˆ theo D y a a ´ Trong d` thi d ˆi x´.ng khˆng khuyˆn, qu´ tr` t` nhˆn l` nhu sau: ¯ˆ ¯o u o o e a ınh ım a a ´ ´ ´ ’ ’ Lˆ y mˆt d ınh bˆ t k` v0 v` d ˇt S0 = {v0 }; sau d o lˆ y mˆt d ınh v1 ∈ Γ(S0 ) v` d at a o ¯˙ a y a ¯a ¯´ a o ¯˙ / a ¯ˇ ´ ´ S1 = S0 ∪ {v1 }; tiˆp theo lˆ y v2 ∈ Γ(S1 ) v` d ˇt S2 = S1 ∪ {v2 }; v` vˆn vˆn... a• b • f • • e • g • d H` 2. 5: ınh 2. 4 ´ ˙ Sˆ ˆ’n d nh ngo`i o o ¯i a - ’ ˙ ’ Dinh ngh˜ 2. 4.1 Cho d` thi G := (V, Γ) Tˆp ho.p con T c´c d ınh cua V l` tˆp ˆ’n d nh ıa ¯o ˆ a a ¯˙ a a o ¯i ˙ ´u moi d ınh khˆng thuˆc T kˆ v´.i ´ nhˆ t mˆt d ınh trong T ; t´.c... 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 H` 2. 8: ınh ˙ ˜ ˜ ´ ’ ’ ˙ ˙ ’ ’ o ¯˙ a o e o ¯˙ e u ta biˆ’u diˆn mˆi v`ng nho bo.i mˆt d ınh v` mˆt canh liˆn thuˆc hai d ınh nˆu hai v`ng e e o u o.ng u.ng c´ chung mˆt biˆn (h˜y v˜ h` a vˆ t` tˆp ˆ’n d inh ngo`i c´ sˆ ˙ ¯ ´ tu ´ o o e a e ınh!) B`i to´n d u ` ım a o a a ¯ e a o o ´ ` ’ ˙ a a ’ ı a a a a ` ¯a a ı a phˆn tu nho nhˆ t-l` β(G) Trong... ˙ ¯o ı: o ¯i 1 V ∈ T 2 T ∈ T v` T ⊂ A th` A ∈ T a ı ´ ˙ ´ ’ o Ta d nh ngh˜ sˆ ˆ’n d nh ngo`i cua d` thi G l` sˆ ¯i ıa o o ¯i a ˙ ¯ˆ a o β(G) := min{#T | T ∈ T } ˙ ˙ ˙ ¯ o ¯˙ ’ e a a o ¯i a ı o o o Tˆp ˆ’n d nh ngo`i cu.c tiˆ’u l` tˆp ˆ’n d nh ngo`i sao cho bo d i mˆt d ınh th` khˆng c`n ˆ’n a o ¯i a ˙ ’ ˙ a d nh ngo`i n˜ ¯i a u ˙ V´ du 2. 4 .2 X´t d` thi G trong H` 2. 5 C´c tˆp {b, g} v` {a,... a o ’ ´ ` ´ - ’ ’ ´ ˙ a ’ liˆn thuˆc hai d ınh nˆu v` chı nˆu hai th`nh phˆn chı kh´c nhau d ung mˆt biˆn D` thi e o ¯˙ e a ˙ e a a ¯´ o e o o.ng u.ng h`m f cho trong H` 2. 12 tu ´ a ınh w xyz • wxyz • e1 e.3 e2 w x yz e4 • e5... f = x y z + xyz + w y 2. 6 2. 6.1 ˙ ¯ˆ ’ Nhˆn cu a d` thi a o ´ C´c d nh l´ vˆ tˆn tai v` duy nhˆ t a ¯i y ` ` e o a a ´ ˙ ¯o ’ ˆ X´t d` thi G := (V, Γ) Ta n´i tˆp ho.p S ⊂ V l` nhˆn cua d` thi G nˆu S l` tˆp ˆ’n d nh e ¯o ˆ o a a a e a a o ¯i ˙ c l` nˆu ´ ’ trong v` ngo`i cua G; t´ a e a a ˙ u ´ 1 Nˆu v ∈ S th` Γ(v) ∩ S = ∅; v` e ı a ´ 2 Nˆu v ∈ S th` Γ(v) ∩ S = ∅ e / ı - ` ` Diˆu kiˆn (1) suy... ngo`i ı e ¯ˆ o ınh a a a a o ¯i a ˙ ˙n d nh ngo`i cu.c tiˆ’u Dˆ thˆ y β(G) = 2 ’ ¯i ˜ a e e ´ C´c tˆp {b, e} v` {a, c, d, f } l` ˆ a a a ao a ˙ ´ ˙ ´ ’ a a V´ du 2. 4.3 V´.i d` thi trong H` 2. 6, tˆp ˆ’n d nh ngo`i cu.c tiˆ’u ch´.a sˆ nho nhˆ t c´c ı o ¯o e u o ˆ ınh a o ¯i a ˙ ` n tu l` {v1 , v4 } v` do d ´ β(G) = 2 ˙ a ’ phˆ a a ¯o ˙ ´ a ˙ ıt a ´ ’ ’ o a a o o ` B`i to´n ta x´t o d ˆy l` xˆy . c´o 92 l`o . i gia ˙’ i nhu . 12 so . d¯ˆo ` du . ´o . i d¯ˆay: ( 726 31485) (61 528 374) (5841 726 3) (35841 726 ) (461 528 37) (5 726 3148) (16837 425 ) (5 726 3184) (4815 726 3) (5146 827 3) ( 427 51863) (3 528 1746) Mˆo ˜ i. v 1 ] − µ 1 [v 2 , v 1 ] + µ[v 2 , v 2 ] + · · · = µ[v 1 , v 1 ] + µ 1 [v 1 , v 2 ] + µ[v 2 , v 2 ] + µ 2 [v 2 , v 3 ] + · · · − (µ 1 + µ 2 + · · · ). Vˆa . y. sˆo ´ . D - i . nh l´y 2. 2.5 [K¨onig] D - ˆo ` thi . vˆo hu . ´o . ng G l`a 2 sˇa ´ c nˆe ´ u v`a chı ˙’ nˆe ´ u n´o khˆong c´o chu tr`ınh c´o d¯ˆo . d`ai le ˙’ . Ch´u . ng minh D - iˆe ` u kiˆe . n