( i ) Tutte多項式

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曲面上のグラフのKrushkal多項式

奈良女子大学大学院人間文化研究科博士前期課程数物科学専攻数学コース

山村瑠納

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3 2

( 2 1

.) 2 1

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１. グラフの基本事項

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)2. p , , V t , , h

( p d

t d

Vv ( r

G

( a biV , r

: 112 o t

, tV x l t

p v_i v_j V t g E tV et

(5)

)2. p , , V t , , h

( p d

t d

Vv ( r

G

( a biV , r

: 112 o t

, tV x l t

112

(6)

)2. p , , V t , , h

( p d

t d

Vv ( r

G

( a biV , r

: 112 o t

, tV x l t

,

112

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)2. p , , V t , , h

( p d

t d

Vv ( r

G

( a biV , r

: 112 o t

, tV x l t

,

et 112

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E

E ) (.

⊆ = ⊆

G .( , ) (.

.( , ) (.

"#$

H

) (.

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( i ) Tutte多項式

2. 背景

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. 8 B 2 BB TV 2 BB +)( - B

1E 1 1 , F 2 BB TV !_" #, % G

!

_"

#, % = ' #

( ) *((")

%

^-())

n 0 E 0 D 0 0 F, 0 , 0

0, 1 4 9

59 4 :

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∖ ℯ ( ∖ e =e D ,/, , : e ∈ $e = v₁ ∪ v₂

,/, , )

e

∖ ℯ ^ℯ

, ,/, , )

ℯ ( _v₁ _~_v₂ ∖ e =e E ) :

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),1- ,.2 G

v ∈ ( v ∈ ( v , v

. ,.2 - ,. _# D ∪ v , v D G

$ = #_&

,.,2, . . ,.2 - ,.

v1 v2

#_$ = #_&

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0 = 8 8 . ,8:: 2 2 8

1A !_" #, % P G a

!_" #, % ) !_"∖' #, % !_"∕' #, %

88 - i 8 ) : = M J GT

!_" #, % )(# 1)ⁱ (% 1) ^j

8 e ∈ , - ( 8 2

: = M 8 M G

!_"_. ₀E ^"_/ #, % ) !_"_. #, % C!_"_/ #, %

1 = 0₃

888 - - - = 8 8 M J GT

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(" 1)

(" 1) (" 1)

(% 1)

(% 1) " 1 % 1 " 1 % 1

e

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1 2

("31)

("31) ("31)

(%31)

(%31) "31 %31 "31 %31

++0+ e 0 0

&

_'

", %

^("31) ^("31) ^(%31) "31 %31 ("31) (%31)

" " " % % " %

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.

l e e h ! ∖ a

i

e : e D

(),., , .

de g ! f G

! ∖ a

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.

l e e h ! ∖ a

i

e : e D

(),., , .

de g ! f G

! ∖ a

!

(18)

.

l e e h ! ∖ a

i

e : e D

(),., , .

de g ! f G

! ∖ a

!

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.

l e e h ! ∖ a

i

e : e D

(),., , .

de g ! f G

! ∖ a

!

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.

l e e h ! ∖ a

i

e : e D

(),., , .

de g ! f G

! ∖ a

!

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EC C G C - 2C 9= 9 . 2 E C (

0 9 9E =E9

!

_"

#, % , !

_"^∗

%, #

0 0MO M; 9 =E9

C9 9 - 2C 9= 9 .E 2 E C 0E9 C J CA 9 9 ; E - :9G C I

GG C CA 9 9E ( ) 7A9G .38 ( (

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( ii ) Krushkal多項式

2. 背景

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i ^KD ^F

F G , ( ,

, ( S:

K , () , , ( ,

⊥ F G , ( ,

F i ^K

( ) (

K i G K i .

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c . i f

C: =e a P_G,Σ, _i X, Y, A, B ^{M g PV}

0 = C76 C 8 7 * 0 0 bd 0 hi

0 0_K⊔ 0 ⊔ GGG ⊔ 0_n 0i i

* 0 ∑ * 0_i

P_G,Σ, _i (

X, Y, A, B

) ∑

X

^c(F)−c(G)

Y

c(Σ F)−c(Σ)^∖

A

^*(S(F))

B

^{*(S (F))}^⊥

- . C

C :

2 C: = . :C = C 6 6 = F C 8 7C , 1 , ( )(

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c . i f

C: =e a P_G,Σ, _i X, Y, A, B ^{M g PV}

0 = C76 C 8 7 * 0 0 bd 0 hi

0 0_K⊔ 0 ⊔ GGG ⊔ 0_n 0i i

* 0 ∑ * 0_i

P_G,Σ, _i (

X, Y, A, B

) ∑

X

^c(F)−c(G)

Y

c(Σ F)−c(Σ)^∖

A

^*(S(F))

B

^{*(S (F))}^⊥

- . C

C :

2 C: = . :C = C 6 6 = F C 8 7C , 1 , ( )(

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,

i

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) ( i

F 1

0 - ) ( - (

- 3∖ ) ( - 3 (

" , ) ( " ,^⊥ ) (

X Y A B

⁽

X B

, )

3∖ ) (

, ^⊥ )

2

(

) X ^c(F)−c(G)Y c(Σ F)−c(Σ)_∖ A "(S(F)) B "(S (F))^⊥

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) ( i

F 1

0 - ) ( - (

- 2∖ ) ( - 2 (

" , ) ( " ,^⊥ ) (

X Y A B

⁽

X

2∖ ) (

, ^⊥ )

(

, )

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) ( i

F 0

- ) ( - (

- 1∖ ) ( - 1 (

" , ) ( " ,^⊥ ) (

X Y A B

⁽

A

, )

1∖ ) (

, ^⊥ )

(

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P_G,Σ, _i ( X, Y, A, B _{) ,} B , B + , , A 1

i

, B , B +10 ,B + 2 , B A 5

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) ( ** * . i i : p be

u ∖i taen G

i oe i oe sh ∖i en i G ld oe f oe D

r c * . : i mG

, ) ( ** * . * .

( ) * , ) i ^： ^{r c}

i ) ( ** * .

"#$

∖ i taen * ) lg

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8 - C 8:1

P_G,Σ, _i (

X, Y, A, B

) P_G*, Σ, i (

Y, X, B, A

)

C 1 2 7 1 8 , i K C1 7 1 8 , P G T

0 - C 8:1 , 1 8 : 1 C1 C 61 2 . 212 C ( )(

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0 G BA (

!_" #, % Y ^((Σ) P_G,Σ, _i

X, Y, Y, Y

^-1

M T 0BAA Q - B 1 b

c , 8 A12 3 B 13 a K TdM

G 1 BA ) B1 8 AB 1 8 A - B 1 81 ) C1 3 8 ) 8 .1A 1A83 (

0 G BA (

!_" #, % Y ^((Σ) P_G,Σ, _i

X, Y, Y, Y

^-1

M T 0BAA Q - B 1 b

c , 8 A12 3 B 13 a K TdM

G 1 BA ) B1 8 AB 1 8 A - B 1 81 ) C1 3 8 ) 8 .1A 1A83 (

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: = ) = 2

P_G,Σ _i ( X, Y, A, B )= P_,∖._,_Σ _i ( X, Y, A, B ) B P_G/e,Σ _i( X, Y, A, B )

e ∈ 1 . 2132 G

e ∈ 132 G

P_G,Σ,_i ( X, Y, A, B )= (X+1 ) P_{G/e, Σ} _i ( X, Y, A, B )

e ∈ : 0 ∖e ( G

P_G,Σ,_i ( X, Y, A, B )= (Y+1 ) P_,∖._{, Σ} _i ( X, Y, A, B )

G P i , . K E

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: = ) = 2

P_G,Σ _i ( X, Y, A, B )= P_,∖._,_Σ _i ( X, Y, A, B ) B P_{G/e,Σ i} ( X, Y, A, B )

e ∈ 1 . 2132 G

e ∈ 132 G

P_G,Σ,_i ( X, Y, A, B )= (X+1 ) P_{G/e, Σ i} ( X, Y, A, B )

e ∈ : 0 ∖e ( G

G P i , . K E

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: = ) = 2

e ∈ 1 . 2132 G

e ∈ 132 G

e ∈ : 0 ∖e ( G

G P i , . K E

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: = ) = 2

e ∈ 1 . 2132 G

e ∈ 132 G

e ∈ : 0 ∖e ( G

G P i , . K E

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3. 結果

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= 2 ,- . / 1,.

) ) = i = ) K

v ∈ ) v ∈ ) ) i G ) i

v , v 2 2 ) _"_# _{= "}_%) i

) ) ) ⋃ ) _"

#~ "_%

"_# = "_%

( ) _" ) ( ) ⋃ ( )

# = "_%

"_#~ "_% ="_#G"_% E

V G a GE

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. G K ()

! ⋂ i , v ⋂ i ∅

D ∖ ⊂ D 1D

v₁

v₂

D 1D i

&_' = &₎

D₂ i²

D i2

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f ∖ . K

⋂ i , ∅ i ⊂ ∖

G K f ^° i . 2

f D1 f ^° i v , f ^° i v

i _& . i ) ,

' = &₎

f ^° i ) ) ∈

f ° i ) ) ∈

i( f⁽

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T T T _!

" = !_$

1 2 1

T T = 1 2 1 , ., . 1. . , . : ,.

i_k = ₍ T₍ K

i = T T

!_" = !_$

K G

P&, (, ) X, Y, A, B P_&_._,(_._,)_. X, Y, A, B P_&₀_,(₀_,)₀ X, Y, A, B

) . .

T T T _!

" = !_$

1 2 1

T T = 1 2 1 , ., . 1. . , . : ,.

i_k = ₍ T₍ K

i = T T

!_" = !_$

K G

P&, (, ) X, Y, A, B P_&_._,(_._,)_. X, Y, A, B P_&₀_,(₀_,)₀ X, Y, A, B

) . .

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: ).

i ′ _j′′ (′ (′′ )11.1, , )

∃

F C ^∀ ( = (′ (′′ )11.1, , ) G

(=(′ (′′

( )11.1, , )

′ (′ )11.1, , ) ′′ (′′ )11.1, , )

i ′ _j′′

i ′ _j′′ (′ (′′ )11.1, , ) G

∀

i ′ _j′′ ( )11.1, , )

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∑ X ^c(F)−c(G)Y c(Σ F)−c(Σ) A ^,(S(F)) B ^{,(S (F))} P_G( X, Y, A, B ) ≝

F ⊆ G

∖ ⊥

(, )

P₆ X, Y, A, B P₆⁷ X, Y, A, B P₆⁷⁷ X, Y, A, B ^F

′ ′′ G

) 1

ij i ′ _j′′ ≤ : ≤ ; , 1 ≤ = ≤ >

) ) .

= C

i ′ _j′′ , ′ ′′ , ) ) .

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∑ X ^c(F)−c(G)Y c(Σ F)−c(Σ) A ^,(S(F)) B ^{,(S (F))} P_G( X, Y, A, B ) ≝

F ⊆ G

∖ ⊥

)(

P₆ X, Y, A, B P₆⁷ X, Y, A, B ,P₆⁷⁷ X, Y, A, B F

ij

P_G( X, Y, A, B ) =

Σ

^R _ij

=

Σ

^R

≤ ; ≤ <

≤ = ≤ >

Σ

^R

≤ ; ≤ < ⁱ′

Σ

^R

≤ = ≤ > ^j′′

. ′ ′′

P₆⁷ X, Y, A, B ,P₆⁷⁷ X, Y, A, B

= G

1 F

i ′ _j′′

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a

, .h b , , h b o n k - . h b J

i l KJ ,- e

, ,- e

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a b

2 1X OI 0 TXEVM II I SEQVMRQ RJ LI 7 XVLNEO SRO[QRPMEO 0 EQGIV MQ 0SSOMI 9E LPE MGV -( ,

6 3OOMV 9RQEKLEQ 2 MIO 9I MQR 4 ESL RO[QRPMEOV EQ LIM 0SSOMGE MRQV I . LI X I SRO[QRPMEO E BM , - CPE L 2; ,6XQ ,

0 2LEPSEQI NE 5 7RJPEQ : RO JXV XEVM II I SEQVMRQ JR LI 1ROOR EV ] MR EQ ] X I SRO[QRPMEO 1XOO 8RQ 9E L ( ) - -,(

A 7 XVLNEO 4 ESLV OMQNV EQ XEOM [ RQ VX JEGIV 2RP MQ R E 2RPSX . ,

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