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PureMathematicsnØêÆ,2021,11(6),1112-1120
PublishedOnlineJune2021inHans.http://www.hanspub.org/journal/pm
https://doi.org/10.12677/pm.2021.116125
r‡:¿ÖãÚdãa
oooûûû
∗
§§§êêê°°°¤¤¤
†
“°¬xŒÆêƆÚOÆ§“°Üw
ÂvFϵ2021c58F¶¹^Fϵ2021c69F¶uÙFϵ2021c616F
Á‡
ü‡ãGÚHÚd…=§‚Ö㊑d.ãGÚ•˜ …=GŠ‘•˜.3ùŸ©
Ù¥,·‚OŽrK
1
∪C
m
(r≥1,m≥3)Š‘dã‡ê,¿• x§Š ‘dãa.Ï
,·‚•OŽrK
1
∪C
m
Údã‡ê,•xrK
1
∪C
m
Údãa.
'…c
Úõ‘ª§Š‘õ‘ª§Úd§Š‘d§Ú•˜§Š‘•˜
TheChromaticEquivalenceClassesofthe
ComplementsofUnionGraphsofr
VerticesandaCycle
DanyangLi
∗
,HaichengMa
†
SchoolofMathematicsandStatistics,QinghaiNationalitiesUniversity,XiningQinghai
Received:May8
th
,2021;accepted:Jun.9
th
,2021;published:Jun.16
th
,2021
∗1˜Šö"
†Ï&Šö"
©ÙÚ^:oû,ê°¤.r‡:¿ÖãÚdãa[J].nØêÆ,2021,11(6):1112-1120.
DOI:10.12677/pm.2021.116125
oû§ê°¤
Abstract
TwographsGandHarechromaticallyequivalentifandonlyifGandHareadjointly
equivalent.GischromaticallyuniqueifandonlyifGadjointlyunique.Inthispaper,
thenumberoftheadjointequivalencegraphsofrK
1
∪C
m
(r≥1,m≥3)iscalculated,
andtheadjointequivalenceclassesofrK
1
∪C
m
canalsobecharacterized.Asaresult,
thenumberofthechromaticequivalencegraphsofrK
1
∪C
m
iscalculated,andthe
chromaticequivalenceclassesofrK
1
∪C
m
canalsobecharacterized.
Keywords
ChromaticPolynomial,AdjointPolynomial,ChromaticallyEquivalent,
AdjointlyEquivalent,ChromaticallyUnique,AdjointlyUnique
Copyright
c
2021byauthor(s)andHansPublishersInc.
This work is licensed undertheCreative Commons Attribution International License (CCBY 4.0).
http://creativecommons.org/licenses/by/4.0/
1.0
©=•Äk•Ã•{ üã.éuãG,G,V(G),E(G),v(G),χ(G)©OL«ãGÖã,:
8,>8,êÚÚê.K
n
,P
n
ÚC
n
(n≥3)©OL«n‡:ã,´Ú,K
1
L«˜‡á
:,D
n
L«C
3
þ˜:†´P
n−2
˜‡à:Å(ã,T
i,j,k
L«•k ˜‡3Ý:,n‡1
Ý:,…ù‡3Ý:n‡1Ý:ål©O•i,j,kä,N
G
(v)L«G¥¤k†:v:
¤8Ü,G∪HL«ãG†ãHØ¿,mGL«m‡GØ¿.
éuêk,XJV(G)˜‡y©{A
1
,A
2
,···,A
k
}¥z‡A
i
´š˜Õá8,Kù‡y©¡
•ãG˜‡k−Õáy©.-α(G,k)L«ãG¤kk−Õáy©ê8,3©[1]¥½ÂãG
Úõ‘ª•
P(G,λ) =
v (G)
P
k=1
α(G,k)(λ)
k
,
ùp(λ)
k
= λ(λ−1)···(λ−k+1),(k≥1).
XJü‡ãGÚH÷vP(G,λ)=P(H,λ),K¡ùü‡ã´Úd,{P•G∼
P
H.² w
/,'X∼
P
´¤kã¤8Ü¥˜‡d'X.†ãGÚd¤kã¤8ÜP•[G]
P
,
DOI:10.12677/pm.2021.1161251113nØêÆ
oû§ê°¤
¡•ãGÚdãa.XJ[G]
P
={G},K¡ ãG´Ú•˜.éu‰½ãG,(½[G]
P
´˜
‡š~k(J¯K.®²kNõã aÚdãa(½(„[2]).•ïÄãÚ5ÚÚõ
‘ª,1987c4V=3[3]¥Ú\Š‘õ‘ªVg,
-P(G,λ) =
v (G)
P
i=1
α(G,i)(λ)
i
•ãGÚõ‘ª,K
h(G,x) =
v (G)
P
i=1
α(G,i)x
i
‰ãGŠ‘õ‘ª.
ãGÚHŠ‘d…=GÚHÚd.-[G]={H|H ∼G}•GŠ‘dã
a.®²uyéõãÚdãa9éõÚ•˜ã[1,4–7],´U•xãÚ
da©Ù„´é.3[5]¥•x8Ü[P
n
]
P
.3[8]¥•x8Ü[aK
1
∪bC
3
S
1≤i≤s
P
l
i
]
P
Ú[aC
3
S
1≤i≤s
P
u
i
S
1≤j≤t
C
v
j
]
P
,ùpa,b´?¿šKê,l
i
´óê,u
i
≥3¿…u
i
6=4(mod5).
±β(G)L«Š‘õ‘ªh(G,x)•¢Š.3[9]¥•xβ(G) >−4¤kã.3ùŸ©Ù¥·
‚•x8Ü[rK
1
∪C
m
],?•x8Ü[rK
1
∪C
m
]
P
.
2.eZÚn
Ún1[8](i)G∼
P
H…=G∼H;
(ii)[G]
P
= {H|H∈[G]};
(iii)ãGÚ•˜…=GŠ‘•˜.
Ún2[1,3]ãGkk‡ëÏ©|µG
1
,G
2
,···G
k
,K
h(G,x) =
k
Y
i=1
h(G
i
,x).
éuãG?¿˜^>e= uv,½ÂãG∗eXe:
V(G∗e) = {V(G)\{u,v}}∪{x},
E(G∗e) = {e∈E(G)|>eØ'é:u½v}∪{xy|y∈N
G
(u)∩N
G
(v)},
Ù¥x/∈V(G).
Ún3[1,10]?¿e∈E(G),h(G) = h(G−e)+h(G∗e),Ù¥G−eL«lãG¥íØ>e.
Ún4[6]G´˜‡ëÏã,Kβ(G) >−4…=
G∈Γ = {K
1
,P
n
(n≥2),T
1,1,k
(k≥1),T
1,2,i
(2 ≤i≤4),C
m
(m≥3),D
l
(4 ≤l≤7)}.
Ún5[9]
(1)P
2m+1
∼P
m
∪C
m+1
(m≥3).
(2)T
1,1,n
∼K
1
∪C
n+2
(n≥2).
DOI:10.12677/pm.2021.1161251114nØêÆ
oû§ê°¤
(3)T
1,2,n
∼K
1
∪D
n+3
.
(4)P
4
∼K
1
∪C
3
.
(5)K
1
∪P
5
∼P
2
∪T
1,1,1
.
(6)C
4
∼D
4
.
(7)P
2
∪C
6
∼P
3
∪D
5
.
(8)P
2
∪C
9
∼P
5
∪D
6
.
(9)K
1
∪C
9
∼T
1,1,1
∪D
6
.
(10)P
2
∪C
15
∼P
5
∪C
5
∪D
7
.
(11)K
1
∪C
15
∼T
1,1,1
∪C
5
∪D
7
.
(12)C
15
∪D
6
∼C
5
∪C
9
∪D
7
.
Ún6[9](1)XJm>n,Kβ(P
m
) <β(P
n
).
(2)β(C
m
) = β(P
2m−1
),Ù¥m≥4.
(3)β(T
1,1,n
) = β(C
n+2
) = β(P
2n+3
),Ù¥n≥2.
(4)β(T
1,2,n
) = β(D
n+3
).
(5)β(C
3
) = β(P
4
).
(6)β(T
1,1,1
) = β(P
5
).
(7)β(C
4
) = β(T
1,1,2
) = β(D
4
) = β(P
7
).
(8)β(C
6
) = β(T
1,1,4
) = β(D
5
) = β(T
1,2,2
) = β(P
11
).
(9)β(C
9
) = β(T
1,1,7
) = β(D
6
) = β(T
1,2,3
) = β(P
17
).
(10)β(C
15
) = β(T
1,1,13
) = β(D
7
) = β(T
1,2,4
) = β(P
29
).
Ún7[6]G´äkβ(G) >−4˜‡ã,KGŠ‘•˜…=
G= kK
1
∪m
2
P
2
∪m
3
P
3
∪m
5
P
5
∪[∪
i≥3
m
2i
P
2i
]∪n
3
C
3
∪[∪
j≥5
n
j
C
j
]∪[∪
5≤l≤7
d
l
D
l
]∪tT
1,1,1
¦
kn
j
= kd
l
= km
5
= m
i
n
i+1
= m
2
n
6
= m
2
n
9
= m
2
n
15
= m
2
t=m
3
d
5
= m
5
d
6
= td
6
= m
5
n
5
d
7
=
tn
5
d
7
= n
15
d
6
= n
5
n
9
d
7
= 0,ùpk,m
i
,n
j
,d
l
,t´šKê.
••B,·‚^δ(G)L«ãG¤kØÓŠ‘dã ‡ê.δ(G)=1… =G´
Š‘•˜.
3.̇(J
½n1êr≥1.(i)m6=3,4,9,15,Kδ(rK
1
∪C
m
)=2,[rK
1
∪C
m
]={rK
1
∪C
m
,(r−
1)K
1
∪T
1,1,m−2
}.
(ii)δ(rK
1
∪C
3
) = 2,[rK
1
∪C
3
] = {rK
1
∪C
3
,(r−1)K
1
∪P
4
}.
DOI:10.12677/pm.2021.1161251115nØêÆ
oû§ê°¤
(iii)δ(rK
1
∪C
4
) = 3,[rK
1
∪C
4
] = {rK
1
∪C
4
,(r−1)K
1
∪T
1,1,2
,rK
1
∪D
4
}.
(iv)
δ(K
1
∪C
9
) =



3,r= 1,
4,r≥2.
[K
1
∪C
9
] =









{K
1
∪C
9
,T
1,1,7
,T
1,1,1
∪D
6
},r= 1,
{rK
1
∪C
9
,(r−1)K
1
∪T
1,1,7
,(r−1)K
1
∪T
1,1,1
∪D
6
,
(r−2)K
1
∪T
1,1,1
∪T
1,2,3
},r≥2.
(v)
δ(K
1
∪C
15
) =









3,r= 1,
5,r= 2,
6,r≥3.
[K
1
∪C
15
] =



























{K
1
∪C
15
,T
1,1,13
,T
1,1,1
∪C
5
∪D
7
},r= 1,
{2K
1
∪C
15
,K
1
∪T
1,1,13
,K
1
∪T
1,1,1
∪C
5
∪D
7
,
T
1,1,1
∪T
1,1,3
∪D
7
,T
1,1,1
∪T
1,2,4
∪C
5
},r= 2,
{rK
1
∪C
15
,(r−1)K
1
∪T
1,1,13
,(r−1)K
1
∪T
1,1,1
∪C
5
∪D
7
,
(r−2)K
1
∪T
1,1,1
∪T
1,1,3
∪D
7
,(r−2)K
1
∪T
1,1,1
∪T
1,2,4
∪C
5
,
(r−3)K
1
∪T
1,1,1
∪T
1,1,3
∪T
1,2,4
},r≥3.
y²:H∼rK
1
∪C
m
,H
1
´H˜‡ëÏ©|,β(H
1
)=β(rK
1
∪C
m
)=β(C
m
),H=
H
1
∪H
2
.
(i)œ/1.m6= 3,4,6,9,15ž,dÚn6•,H
1
= C
m
,T
1,1,m−2
,P
2m−1
.
H
1
= C
m
ž, drK
1
∪C
m
∼C
m
∪H
2
H
2
∼rK
1
, dÚn7 •rK
1
Š‘•˜, KH
2
= rK
1
.
H
1
= T
1,1,m−2
ž,dT
1,1,m−2
∪H
2
∼rK
1
∪C
m
∼(r−1)K
1
∪T
1,1,m−2
H
2
∼(r−1)K
1
,
?˜ÚH
2
= (r−1)K
1
.
H
1
= P
2m−1
ž,drK
1
∪C
m
∼P
2m−1
∪H
2
∼P
m−1
∪C
m
∪H
2
rK
1
∼P
m−1
∪H
2
,dÚ
n7•rK
1
Š‘•˜,KùH
2
´Ø•3.
m6= 3,4,6,9,15ž,Hkü‡:rK
1
∪C
m
,(r−1)K
1
∪T
1,1,m−2
.
œ/2.m= 6ž,dÚn6•,H
1
= C
6
,T
1,1,4
,D
5
,T
1,2,2
,P
11
.
H
1
= C
6
ž,drK
1
∪C
6
∼C
6
∪H
2
H
2
= rK
1
.
H
1
= T
1,1,4
ž,drK
1
∪C
6
∼T
1,1,4
∪H
2
∼K
1
∪C
6
∪H
2
H
2
= (r−1)K
1
.
DOI:10.12677/pm.2021.1161251116nØêÆ
oû§ê°¤
H
1
= D
5
ž,drK
1
∪C
6
∼D
5
∪H
2
,dÚn5(7)?˜ÚP
3
∪rK
1
∪C
6
∼P
3
∪D
5
∪H
2
∼
P
2
∪C
6
∪H
2
P
3
∪rK
1
∼P
2
∪H
2
,dÚn7•P
3
∪rK
1
Š‘•˜,¤±ùH
2
´Ø•3.
H
1
= T
1,2,2
ž,drK
1
∪C
6
∼T
1,2,2
∪H
2
∼K
1
∪D
5
∪H
2
,dÚn5(7)?˜ÚP
3
∪rK
1
∪
C
6
∼P
3
∪D
5
∪K
1
∪H
2
∼P
2
∪C
6
∪K
1
∪H
2
P
3
∪(r−1)K
1
∼P
2
∪H
2
,dÚn7•P
3
∪(r−1)K
1
Š‘•˜,¤±ùH
2
´Ø•3.
H
1
=P
11
ž,drK
1
∪C
6
∼P
11
∪H
2
∼P
5
∪C
6
∪H
2
rK
1
∼P
5
∪H
2
,dÚn7•rK
1
Š‘•˜,¤±ùH
2
´Ø•3.
m= 6ž,Hkü‡:rK
1
∪C
6
,(r−1)K
1
∪T
1,1,4
.
nþ¤ã,m6= 3,4,9,15ž,Hkü‡:rK
1
∪C
m
,(r−1)K
1
∪T
1,1,m−2
.
(ii)m= 3ž,dÚn6•,H
1
= C
3
,P
4
.
H
1
= C
3
ž,drK
1
∪C
3
∼C
3
∪H
2
H
2
= rK
1
.
H
1
= P
4
ž,drK
1
∪C
3
∼P
4
∪H
2
∼K
1
∪C
3
∪H
2
H
2
= (r−1)K
1
.
m= 3ž,Hkü‡:rK
1
∪C
3
,(r−1)K
1
∪P
4
.
(iii)m= 4ž,dÚn6•,H
1
= C
4
,T
1,1,2
,D
4
,P
7
.
H
1
= C
4
ž,drK
1
∪C
4
∼C
4
∪H
2
H
2
= rK
1
.
H
1
= T
1,1,2
ž,drK
1
∪C
4
∼T
1,1,2
∪H
2
∼K
1
∪C
4
∪H
2
H
2
= (r−1)K
1
.
H
1
= D
4
ž,drK
1
∪C
4
∼D
4
∪H
2
∼C
4
∪H
2
H
2
= rK
1
.
H
1
= P
7
ž,drK
1
∪C
4
∼P
7
∪H
2
∼P
3
∪C
4
∪H
2
rK
1
∼P
3
∪H
2
,dÚn7•rK
1
Š
‘•˜,KùH
2
´Ø•3.
m= 4ž,Hkn‡:rK
1
∪C
4
,(r−1)K
1
∪T
1,1,2
,rK
1
∪D
4
.
(iv)m= 9ž,dÚn6•,H
1
= C
9
,T
1,1,7
,D
6
,T
1,2,3
,P
17
.
œ/1.r= 1ž:
H
1
= C
9
ž,dK
1
∪C
9
∼C
9
∪H
2
H
2
= K
1
.
H
1
= T
1,1,7
ž,dK
1
∪C
9
∼T
1,1,7
∪H
2
∼K
1
∪C
9
∪H
2
H
2
•˜ã.
H
1
=D
6
ž,dD
6
∪H
2
∼K
1
∪C
9
∼T
1,1,1
∪D
6
H
2
∼T
1,1,1
,dÚn7•T
1,1,1
Š‘•
˜,KH
2
= T
1,1,1
.
H
1
=T
1,2,3
ž,dK
1
∪C
9
∼T
1,2,3
∪H
2
∼K
1
∪D
6
∪H
2
C
9
∼D
6
∪H
2
,dÚn7•C
9
Š‘•˜,KùH
2
´Ø•3.
H
1
= P
17
ž,dK
1
∪C
9
∼P
17
∪H
2
∼P
8
∪C
9
∪H
2
K
1
∼P
8
∪H
2
,dÚn7•K
1
Š‘
•˜,KùH
2
´Ø•3.
m= 9(r= 1)ž,Hkn‡:K
1
∪C
9
,T
1,1,7
,T
1,1,1
∪D
6
.
œ/2.r= 2ž,†œ/1aq,Ñ.
DOI:10.12677/pm.2021.1161251117nØêÆ
oû§ê°¤
œ/3.r≥3ž:
H
1
= C
9
ž,drK
1
∪C
9
∼C
9
∪H
2
H
2
= rK
1
.
H
1
= T
1,1,7
ž,drK
1
∪C
9
∼T
1,1,7
∪H
2
∼K
1
∪C
9
∪H
2
H
2
= (r−1)K
1
.
H
1
= D
6
ž,dD
6
∪H
2
∼rK
1
∪C
9
∼(r−1)K
1
∪T
1,1,1
∪D
6
H
2
∼(r−1)K
1
∪T
1,1,1
,
dÚn7•(r−1)K
1
∪T
1,1,1
Š‘•˜,KH
2
= (r−1)K
1
∪T
1,1,1
.
H
1
= T
1,2,3
ž,dT
1,2,3
∪H
2
∼rK
1
∪C
9
∼(r−1)K
1
∪T
1,1,1
∪D
6
∼(r−2)K
1
∪T
1,1,1
∪
T
1,2,3
H
2
∼(r−2)K
1
∪T
1,1,1
,dÚn7•((r−2)K
1
∪T
1,1,1
Š‘•˜,KH
2
= (r−2)K
1
∪T
1,1,1
.
H
1
=P
17
ž,drK
1
∪C
9
∼P
17
∪H
2
∼P
8
∪C
9
∪H
2
rK
1
∼P
8
∪H
2
,dÚn7•rK
1
Š‘•˜,KùH
2
´Ø•3.
m= 9 ž,Hko‡: rK
1
∪C
9
,(r−1)K
1
∪T
1,1,7
,(r−1)K
1
∪T
1,1,1
∪D
6
,(r−2)K
1
∪T
1,1,1
∪
T
1,2,3
.
(v)m= 15ž,dÚn6•,H
1
= C
15
,T
1,1,13
,D
7
,T
1,2,4
,P
29
.
œ/1.r= 1ž:
H
1
= C
15
ž,dK
1
∪C
15
∼C
15
∪H
2
H
2
= K
1
.
H
1
= T
1,1,13
ž,dK
1
∪C
15
∼T
1,1,13
∪H
2
∼K
1
∪C
15
∪H
2
H
2
•˜ã.
H
1
=D
7
ž,dD
7
∪H
2
∼K
1
∪C
15
∼T
1,1,1
∪C
5
∪D
7
H
2
∼T
1,1,1
∪C
5
,dÚn7
•T
1,1,1
∪C
5
Š‘•˜,KH
2
= T
1,1,1
∪C
5
..
H
1
= T
1,2,4
ž,dK
1
∪C
15
∼T
1,2,4
∪H
2
∼K
1
∪D
7
∪H
2
C
15
∼D
7
∪H
2
,dÚn7 •C
15
Š‘•˜,KùH
2
´Ø•3.
H
1
=P
29
ž,dK
1
∪C
15
∼P
29
∪H
2
∼P
14
∪C
15
∪H
2
K
1
∼P
14
∪H
2
,dÚn7•K
1
Š‘•˜,KùH
2
´Ø•3.
m= 15(r= 1)ž,Hkn‡:K
1
∪C
15
,T
1,1,13
,T
1,1,1
∪C
5
∪D
7
.
œ/2.r= 2ž:
H
1
= C
15
ž,d2K
1
∪C
15
∼C
15
∪H
2
H
2
= 2K
1
.
H
1
= T
1,1,13
ž,d2K
1
∪C
15
∼T
1,1,13
∪H
2
∼K
1
∪C
15
∪H
2
H
2
= K
1
.
H
1
= D
7
ž,dD
7
∪H
2
∼2K
1
∪C
15
∼K
1
∪T
1,1,1
∪C
5
∪D
7
H
2
∼K
1
∪T
1,1,1
∪C
5
,d
Ún5•K
1
∪T
1,1,1
∪C
5
∼T
1,1,1
∪T
1,1,3
,?˜ÚdÚn7•H
2
= K
1
∪T
1,1,1
∪C
5
,T
1,1,1
∪T
1,1,3
.
H
1
=T
1,2,4
ž,dT
1,2,4
∪H
2
∼2K
1
∪C
15
∼K
1
∪T
1,1,1
∪C
5
∪D
7
∼T
1,1,1
∪C
5
∪T
1,2,4
H
2
∼T
1,1,1
∪C
5
,dÚn7•T
1,1,1
∪C
5
Š‘•˜,KH
2
= T
1,1,1
∪C
5
.
H
1
= P
29
ž,d2K
1
∪C
15
∼P
29
∪H
2
∼P
14
∪C
15
∪H
2
2K
1
∼P
14
∪H
2
,dÚn7•2K
1
Š‘•˜,KùH
2
´Ø•3.
m= 15(r= 2)ž,Hkʇ:2K
1
∪C
15
,K
1
∪T
1,1,13
,K
1
∪T
1,1,1
∪C
5
∪D
7
,T
1,1,1
∪T
1,1,3
∪
D
7
,T
1,1,1
∪C
5
∪T
1,2,4
.
DOI:10.12677/pm.2021.1161251118nØêÆ
oû§ê°¤
œ/3.r= 3ž,†œ/2aq,Ñ.
œ/4.r≥4ž:
H
1
= C
15
ž,drK
1
∪C
15
∼C
15
∪H
2
H
2
= rK
1
.
H
1
= T
1,1,13
ž,drK
1
∪C
15
∼T
1,1,13
∪H
2
∼K
1
∪C
15
∪H
2
H
2
= (r−1)K
1
.
H
1
=D
7
ž,dD
7
∪H
2
∼rK
1
∪C
15
∼(r−1)K
1
∪T
1,1,1
∪C
5
∪D
7
H
2
∼(r−
1)K
1
∪T
1,1,1
∪C
5
,dÚn5•(r−1)K
1
∪T
1,1,1
∪C
5
∼(r−2)K
1
∪T
1,1,1
∪T
1,1,3
,?˜ÚdÚn7
•H
2
= (r−1)K
1
∪T
1,1,1
∪C
5
,(r−2)K
1
∪T
1,1,1
∪T
1,1,3
.
H
1
= T
1,2,4
ž, dT
1,2,4
∪H
2
∼rK
1
∪C
15
∼(r−1)K
1
∪T
1,1,1
∪C
5
∪D
7
∼(r−2)K
1
∪T
1,1,1
∪
C
5
∪T
1,2,4
H
2
∼(r−2)K
1
∪T
1,1,1
∪C
5
, dÚn5 •(r−2)K
1
∪T
1,1,1
∪C
5
∼(r−3)K
1
∪T
1,1,1
∪T
1,1,3
,
?˜ÚdÚn7•H
2
= (r−2)K
1
∪T
1,1,1
∪C
5
,(r−3)K
1
∪T
1,1,1
∪T
1,1,3
.
H
1
= P
29
ž,drK
1
∪C
15
∼P
29
∪H
2
∼P
14
∪C
15
∪H
2
rK
1
∼P
14
∪H
2
,dÚn7•rK
1
Š‘•˜,KùH
2
´Ø•3.
m=15ž,Hk8‡µrK
1
∪C
15
,(r−1)K
1
∪T
1,1,13
,(r−1)K
1
∪T
1,1,1
∪C
5
∪D
7
,(r−
2)K
1
∪T
1,1,1
∪T
1,1,3
∪D
7
,(r−2)K
1
∪T
1,1,1
∪C
5
∪T
1,2,4
,(r−3)K
1
∪T
1,1,1
∪T
1,1,3
∪T
1,2,4
.
íØ1éu½n1¥mØÓa.,[
rK
1
∪C
m
]
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8Ü.
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ë•©z
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[2]Dong,F.M.,Koh,K.M.andTeo,K.T.(2005)ChromaticPolynomialsandChromaticityof
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1≤m
0
lm
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−1(λ)
l
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oû§ê°¤
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