Các số cơ bản của đồ thị

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

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

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

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
•
•
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.
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

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

• 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

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

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

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2
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5

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6
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• • •
•
•
•
•
•
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

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 alm 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