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AdvancesinAppliedMathematics
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,2023,12(6),3030-3038
PublishedOnlineJune2023inHans.https://www.hanspub.org/journal/aam
https://doi.org/10.12677/aam.2023.126304
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TheAnti-RamseyNumber
ofUnconnectedGraphsin
PlaneTriangulationGraphs
DonglianLuo
∗
,JunqiGu
SchoolofMathematicalSciences,ZhejiangNormalUniversity,JinhuaZhejiang
Received:May28
th
,2023;accepted:Jun.23
rd
,2023;published:Jun.30
th
,2023
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3030-3038.DOI:10.12677/aam.2023.126304
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Abstract
Givenanedge-coloringofagraph
G
,
G
issaidtobe
rainbow
ifanytwoedgesof
G
receivedifferentcolors.Giventwographs
G
and
H
,theanti-Ramseynumberof
H
in
G
isdefinedtobethemaximumnumberofcolorsinanedge-coloredgraph
G
which
containsnorainbowcopiesof
H
.Theanti-Ramseynumbersforgraphs,especially
matchings,havebeenstudiedinseveralgraphclasses.GilboaandRodittyfocused
ontheanti-Ramseynumberofgraphswithsmallcomponents,buttheresultsofthe
anti-Ramsey number ofgraphswithsmallcomponents inplane grapharefew.Inthis
paper,wecontinuetheworkinthisdirectanddeterminetheanti-Ramseynumberof
C
3
∪
tP
2
in plane triangulations,thenwe can get
2
n
+3
t
−
9
≤
AR
(
T
n
,C
3
∪
tP
2
)
≤
2
n
+4
t
−
5
for
n
≥
2
t
+3
,t
≥
2
.
Keywords
RainbowMatching,Anti-RamseyNumber,PlaneTriangulations
Copyright
c
2023byauthor(s)andHansPublishersInc.
This work is licensed undertheCreative Commons Attribution InternationalLicense(CCBY4.0).
http://creativecommons.org/licenses/by/4.0/
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DOI:10.12677/aam.2023.1263043032
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−
V
1
X
„
ãã
2
¤
«
.
Ø
J
u
y
,
3
ã
G
−
V
1
¥
k
t
‡
o
¡
.
d
ž
d
u
vw/
∈
E
(
T
n
),
Ï
d
3
ã
G
−
V
1
¥
z
‡
o
¡
S
–
k
1
‡
Ý
ê
u
2
:
.
b
•
3
º:
x
∈
V
1
3
v
1
v
2
vwv
1
Œ
¤
o
¡
S
…
÷
v
d
(
x
) =1.
Ø
”
b
v
1
x
∈
E
(
G
).
Ø
J
u
y
,
3
ã
T
n
−
v
1
v
2
¥
¤
k
>
ô
Ú
þ
•
c
(
v
1
v
2
).
Ï
d
f
ã
T
n
[
v
1
,x,w
]
∪
T
n
[
v
2
,v
3
]
´
˜
‡
ç
ô
C
3
∪
P
2
,
g
ñ
.
l
Œ
•
é
?
¿
3
o
¡
S
º:
x
∈
V
1
Ñ
k
d
(
x
) = 0.
Figure2.
G
−
V
1
ã
2.
G
−
V
1
Ï
d
·
‚
Œ
±
e
(
G
)
≤
3+2
t
+(
n
−
3
−
2
t
) =
n
.
n
= 2
k
ž
,
k
e
(
G
) =
b
3
n
−
4
2
c
= 3
k
−
2.
Ï
d
3
k
−
2
≤
n
=2
k
.
l
k
k
≤
2,
g
ñ
.
n
=2
k
+ 1
ž
,
k
e
(
G
)=
3
n
−
4
2
=3
k
.
Ï
d
DOI:10.12677/aam.2023.1263043033
A^
ê
Æ
?
Ð
Û
Á
ë
§
d
j
3
k
≤
n
=2
k
+ 1.
l
Œ
±
k
≤
1,
g
ñ
.
Ï
d
ã
T
n
?
¿
3
n
−
4
2
-
>
/Ú
•
¹
˜
‡
ç
ô
C
3
∪
P
2
,
=
AR
(
T
n
,C
3
∪
P
2
)
≤
3
n
−
4
2
−
1.
n
þ
¤
ã
,
½
n
2
y
²
.
.
3.
²
¡
n
¿
©
ã
¥
C
3
∪
tP
2
Anti-Ramsey
ê
½
n
3.
é
?
¿
n
≥
2
t
+3
,t
≥
2
,
AR
(
T
n
,C
3
∪
tP
2
)
≥
2
n
+3
t
−
9
.
y
²
:
-
P
´
º:
^
S
•
v
1
,v
2
,
···
,v
t
−
2
˜
^
´
.
H
´
Ï
L
‰
P
O
\
ü
‡
ƒ
ë
º:
x,y
¿
…
r
x
Ú
y
ü
‡
:
†
P
þ
z
‡
:
ƒ
ë
²
¡
n
¿
©
ã
,
H
•
>
²
¡
±
x,y,v
1
Š
•
>
.
.
Ï
d
k
|
H
|
=
t
.
-
T
H
´
Ï
L
‰
H
z
‡
¡
F
O
\
˜
‡
#
º:
,
ò
Ù
†
3
F
¥
¤
k
º:
ƒ
ë
¤
²
¡
n
¿
©
ã
.
Ï
d
T
H
´
º:
ê
•
t
+ (2
t
−
4)=3
t
−
4
n
¿
©
ã
.
-
w
´
O
\
3
H
L
¡
˜
‡
#
º:
.
-
T
´
Ï
L
‰
T
H
•
¹
x,y,w
¡
O
\
n
−
(3
t
−
4)
‡
º:
,
Ù
¥
O
\
º:
•
w
1
,w
2
,
···
,w
n
−
3
t
+4
,
¦
{
ww
1
,w
n
−
3
t
+4
x,w
n
−
3
t
+4
y
}⊆
E
(
T
)
…
é
¤
k
i
∈{
1
,
···
,n
−
3
t
+ 3
}
ò
w
i
†
T
¥
x,y,w
i
+1
ƒ
ë
¤
n
‡
º:
²
¡
n
¿
©
ã
.
t
= 4
…
n
= 12
ž
ã
T
E
X
„
ãã
3
¤
«
.
Figure3.
Theconstructionof
T
,
t
=4and
n
=12
ã
3.
t
=4
…
n
=12
ž
ã
T
E
w
,
,
T
∈T
n
.
-
c
´
T
˜
‡
>
/Ú
,
Ä
k
‰
>
ww
1
,ww
2
,
···
,w
n
−
3
t
+3
w
n
−
3
t
+4
/Ú
•
1,
,
‰
¤
k
T
•
e
>
^†
1
Ø
Ó
…
p
Ø
ƒ
Ó
ô
Ú/Ú
.
d
ž
Œ
±
N
´
u
y
T
Ø
•
¹
ç
ô
C
3
∪
tP
2
…
c
ô
Ú
ê´
(3
n
−
6)
−
(
n
−
3
t
+4)+1 = 2
n
+3
t
−
9.
Ï
d
y
²
AR
(
T
n
,C
3
∪
tP
2
)
≥
2
n
+3
t
−
9.
n
þ
¤
ã
,
½
n
3
y
²
.
.
½
n
4.
é
?
¿
ê
n
≥
2
t
+3
,t
≥
2
,
AR
(
T
n
,C
3
∪
tP
2
)
≤
2
n
+4
t
−
5
.
y
²
:
·
‚
Ï
L
é
t
?
1
8
B
5
y
²
.
t
=2
ž
,
é
?
¿
n
≥
7,
y
²
Ø
ª
AR
(
T
n
,C
3
∪
2
P
2
)
≤
2
n
+3.
·
‚
•
I
y
²
?
¿
ã
T
n
∈T
n
?
¿
(2
n
+4)-
>
/Ú
Ñ
•
¹
˜
‡
ç
ô
C
3
∪
2
P
2
.
‡
L
5
,
·
‚
b
•
3
˜
‡
ã
T
n
(2
n
+4)-
>
/Ú
Ø
•
¹
ç
ô
C
3
∪
2
P
2
.
Š
â
½
n
2,
Œ
±
N
´
ã
T
n
DOI:10.12677/aam.2023.1263043034
A^
ê
Æ
?
Ð
Û
Á
ë
§
d
j
•
¹
˜
‡
d
H
1
Ú
H
2
Ø
ƒ
¿
|
¤
ç
ô
f
ã
H
,
Ù
¥
H
1
=
C
3
…
H
2
=
P
2
.
-
ã
G
´
ã
T
n
˜
‡
>
ê
•
2
n
+4
ç
ô
)
¤
f
ã
…
•
¹
f
ã
H
.
-
V
(
H
1
) =
{
v
1
,v
2
,v
3
}
,
V
(
H
2
) =
{
v
4
,v
5
}
…
D
=
V
(
T
n
)
\
V
(
H
).
é
?
¿
º:
v
∈
V
(
G
),
^
d
(
v
)
L
«
3
ã
G
¥
:
v
Ý
.
w
,
,
ã
G
Ø
•
¹
†
ã
C
3
∪
2
P
2
Ó
f
ã
.
Ï
d
é
?
¿
º:
v,w
∈
D
k
vw/
∈
E
(
G
).
l
é
?
¿
v
∈
D
k
0
≤
d
(
v
)
≤
5.
e
5
éº:
v
∈
D
Š
â
d
(
v
)
?
1
©
œ
¹
?
Ø
.
œ
/
1.
•
3
˜
‡
º:
v
∈
D
¦
d
(
v
) = 5.
d
ž
{
vv
1
,vv
2
,vv
3
,vv
4
,vv
5
}⊆
E
(
G
).
X
J
•
3
º:
w
∈
D
\{
v
}
Ú
i
∈{
1
,
···
,
5
}
¦
wv
i
∈
E
(
G
),
@
o
Œ
±
N
´
é
˜
‡
ç
ô
C
3
∪
2
P
2
.
Ï
d
Œ
±
é
?
¿
º:
w
∈
D
\{
v
}
Ñ
k
d
(
w
) = 0.
l
2
n
+4 =
e
(
G
) =
e
(
G
[
V
(
H
)])+
e
G
(
V
(
H
)
,D
)
≤
e
(
T
5
)+5 = 14.
Œ
±
n
≤
5,
g
ñ
.
œ
/
2.
•
3
˜
‡
º:
v
∈
D
¦
d
(
v
) = 4.
b
{
vv
1
,vv
2
,vv
3
,vv
5
}⊆
E
(
G
).
X
J
•
3
º:
w
∈
D
\{
v
}
Ú
i
∈{
1
,
···
,
4
}
¦
wv
i
∈
E
(
G
),
@
o
N
´
é
˜
‡
ç
ô
C
3
∪
2
P
2
.
Ï
d
Œ
±
é
?
¿
w
∈
D
\{
v
}
Ñ
k
d
(
w
)
≤
1.
l
2
n
+4 =
e
(
G
) =
e
(
G
[
V
(
H
)])+
e
G
(
V
(
H
)
,D
)
≤
e
(
T
5
)+(
n
−
6)+4 =
n
+7.
Œ
±
n
≤
3,
g
ñ
.
b
{
vv
1
,vv
2
,vv
4
,vv
5
}⊆
E
(
G
).
X
J
•
3
º:
w
∈
D
\{
v
}
Ú
i
∈{
3
,
4
,
5
}
¦
wv
i
∈
E
(
G
),
@
o
N
´
é
˜
‡
ç
ô
C
3
∪
2
P
2
.
X
J
•
3
º:
w
∈
D
\{
v
}
¦
wv
1
∈
E
(
G
),
@
o
·
‚
Œ
±
u
y
G
[
v,v
4
,v
5
]
∪
G
[
w,v
1
]
∪
G
[
v
2
,v
3
]
´
˜
‡
ç
ô
C
3
∪
2
P
2
,
g
ñ
.
X
J
•
3
º:
w
∈
D
\{
v
}
¦
wv
2
∈
E
(
G
),
@
o
Œ
±
u
y
G
[
v,v
4
,v
5
]
∪
G
[
w,v
2
]
∪
G
[
v
1
,v
3
]
´
˜
‡
ç
ô
C
3
∪
2
P
2
,
g
ñ
.
Ï
d
Œ
±
é
?
¿
º:
w
∈
D
\{
v
}
Ñ
k
d
(
w
) =0.
l
2
n
+4=
e
(
G
)=
e
(
G
[
V
(
H
)])+
e
G
(
V
(
H
)
,D
)
≤
e
(
T
5
)+4 = 13.
Œ
±
n<
4,
g
ñ
.
œ
/
3.
•
3
˜
‡
º:
v
∈
D
¦
d
(
v
) = 3.
b
{
vv
1
,vv
2
,vv
3
}⊆
E
(
G
).
X
J
•
3
º:
w
∈
D
\{
v
}
Ú
i
∈{
1
,
2
,
3
}
¦
wv
i
∈
E
(
G
),
@
o
N
´
é
˜
‡
ç
ô
C
3
∪
2
P
2
,
g
ñ
.
Ï
d
Œ
±
é
?
¿
º:
w
∈
D
\{
v
}
Ñ
k
d
(
w
)
≤
2.
l
2
n
+4 =
e
(
G
) =
e
(
G
[
V
(
H
)])+
e
G
(
V
(
H
)
,D
)
≤
e
(
T
5
)+2(
n
−
6)+3 = 2
n
,
g
ñ
.
b
{
vv
2
,vv
4
,vv
5
}⊆
E
(
G
).
X
J
•
3
º:
w
∈
D
\{
v
}
Ú
i
∈{
1
,
···
,
5
}
¦
wv
i
∈
E
(
G
),
@
o
N
´
é
˜
‡
ç
ô
C
3
∪
2
P
2
.
Ï
d
Œ
±
é
?
¿
º:
w
∈
D
\{
v
}
Ñ
k
d
(
w
)=0.
l
2
n
+4 =
e
(
G
) =
e
(
G
[
V
(
H
)])+
e
G
(
V
(
H
)
,D
)
≤
e
(
T
5
)+3 = 12.
Œ
±
n
≤
4,
g
ñ
.
b
{
vv
1
,vv
2
,vv
5
}⊆
E
(
G
).
X
J
•
3
º:
w
∈
D
\{
v
}
Ú
i
∈{
3
,
4
}
¦
wv
i
∈
E
(
G
),
@
o
N
´
é
˜
‡
ç
ô
C
3
∪
2
P
2
.
b
•
3
º:
w
∈
D
\{
v
}
¦
{
wv
1
,wv
2
,wv
5
}⊆
E
(
G
).
•
Ä
f
ã
T
n
[
v
1
,v
2
,v
3
]
∪
T
n
[
v,v
4
]
∪
T
n
[
w,v
5
],
f
ã
T
n
[
v,v
4
,v
5
]
∪
T
n
[
v
2
,v
3
]
∪
T
n
[
w,v
1
],
±
9
f
ã
T
n
[
v,v
4
,v
5
]
∪
T
n
[
v
1
,v
3
]
∪
T
n
[
w,v
2
],
Œ
±
c
(
vv
4
)
∈{
v
1
v
2
,v
1
v
3
,v
2
v
3
,wv
5
}∩{
v
4
v
5
,vv
5
,v
2
v
3
,wv
1
}∩
{
v
4
v
5
,vv
5
,v
1
v
3
,wv
2
}
=
∅
,
g
ñ
.
Ï
d
é
?
¿
º:
w
∈
D
\{
v
}
Ñ
k
d
(
w
)
≤
2.
l
2
n
+4 =
e
(
G
) =
e
(
G
[
V
(
H
)])+
e
G
(
V
(
H
)
,D
)
≤
e
(
T
5
)+2(
n
−
6)+3 = 2
n
,
g
ñ
.
DOI:10.12677/aam.2023.1263043035
A^
ê
Æ
?
Ð
Û
Á
ë
§
d
j
œ
/
4.
z
‡
º:
v
∈
D
Ñ
k
d
(
v
)
≤
2.
d
ž
2
n
+4=
e
(
G
)=
e
(
G
[
V
(
H
)])+
e
G
(
V
(
H
)
,D
)
≤
e
(
T
5
)+2(
n
−
5) =2
n
−
1,
g
ñ
.
Ï
d
é
?
¿
n
≥
7,
AR
(
T
n
,C
3
∪
2
P
2
)
≤
2
n
+3.
t
= 2
ž
½
n
4
y
²
.
.
-
t
≥
3.
e
5
y
²
é
?
¿
n
≥
2
t
+ 3
k
AR
(
T
n
,C
3
∪
tP
2
)
≤
2
n
+4
t
−
5.
·
‚
•
I
‡
y
²
?
¿
ã
T
n
∈T
n
?
¿
(2
n
+4
t
−
4)-
>
/Ú
Ñ
•
¹
˜
‡
ç
ô
C
3
∪
tP
2
.
‡
L
5
,
·
‚
b
•
3
˜
‡
ã
T
n
(2
n
+4
t
−
4)-
>
/Ú
¦
ã
T
n
Ø
•
¹
ç
ô
C
3
∪
tP
2
.
Š
â
8
B
b
,
Œ
±
N
´
ã
T
n
•
¹
˜
‡
d
H
1
Ú
H
2
Ø
ƒ
¿
|
¤
ç
ô
f
ã
H
,
Ù
¥
H
1
=
C
3
…
H
2
=(
t
−
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