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AdvancesinAppliedMathematics
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,2022,11(1),238-245
PublishedOnlineJanuary2022inHans.http://www.hanspub.org/journal/aam
https://doi.org/10.12677/aam.2022.111030
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Anti-Ramsey
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RainbowMatchingsinPlanarGraphs
RuiYu
1
,HuapingWang
2
1
CollegeofMathematicsandComputerScience,ZhejiangNormalUniversity,JinhuaZhejiang
2
SchoolofMathematicsandStatistics,JiangxiNormalUniversity,NanchangJiangxi
Received:Dec.18
th
,2021;accepted:Jan.7
th
,2022;published:Jan.21
st
,2022
Abstract
Wecallanedge-coloredgraph
G
rainbow,ifallofitsedgeshavedifferentcolors.
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A^
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,2022,11(1):238-245.
DOI:10.12677/aam.2022.111030
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Theanti-Ramseynumberofthegraph
H
in
G
,denotedby
AR
(
G,H
)
,isthemaximum
number ofcolors inan edge-coloringof
G
whichdoes notcontainanyrainbowsubgraph
isomorphicto
H
.Inthispaper,weconsidertheanti-Ramseynumberformatchings
inplanargraphs
T
−
n
.
Keywords
Matching,RainbowGraph,Anti-RamseyNumber
Copyright
c
2022byauthor(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.2022.111030239
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T
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ñ
.
DOI:10.12677/aam.2022.111030240
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Figure3.
s
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1
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3.
s
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1
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e
s
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q
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1
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8,
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2).
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8
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1
v
i
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1
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i
|
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≤
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≤
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(
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,
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n
2.1,
•
3
V
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f
8
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S
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4.
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=
s
,
o
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G
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)
−|
S
|
=
q
.
-
G
−
S
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•
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1
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1
,
···
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q
,
-
|
A
i
|
=
a
i
(1
≤
i
≤
q
).
Ø
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˜
„
5
,
·
‚
a
1
≥
a
2
≥···≥
a
q
≥
1.
-
G
−
S
ó
©
|
•
C
(
G
) =
G
−
q
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i
=1
V
(
A
i
)
∪
S
.
du
q
=
s
+
n
−
4
…
s
+
q
≤
n
,
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d
0
≤
s
≤
2.
e
s
=0,
q
=
n
−
4.
a
1
≤
3
ž
,
k
|
E
(
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)
|≤
6
<n
−
1,
†
b
g
ñ
.
Ï
d
,
C
(
G
)=
∅
,
a
1
=5,
a
2
=
a
2
=
···
=
a
q
=1,
|
E
(
G
)
|≤
3
×
5
−
7=8.
¤
±
,
n
=7.
-
V
(
A
1
)=
{
u
1
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2
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3
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4
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5
}
,
V
(
A
2
)=
{
v
2
}
,
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(
A
3
)=
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±
3
V
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3
3
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v
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v
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1
)
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+
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2
)
|
+
|
E
(
A
1
,v
3
)
|
=
|
E
(
T
−
7
)
|−|
E
(
A
1
)
|
= 6.
©
Û
Œ
,
3
T
−
7
¥
,
{
v
2
,v
3
}
†
u
i
(
i
= 1
,
2
,
3)
˜
½
ƒ
.
Ø
”
u
1
v
2
∈
T
−
7
,
®
•
A
1
−
u
1
•
¹
˜
‡
DOI:10.12677/aam.2022.111030241
A^
ê
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Ð
{
b
§
u
²
ç
ô
o
,
d
Ú
n
2.2
Œ
T
−
7
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•
3
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‡
ç
ô
3
K
2
,
†
b
g
ñ
.
e
s
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q
=
n
−
3.
X
J
|
C
(
G
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=2,
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o
|
E
(
G
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n
−
1 +1=
n<n
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†
b
g
ñ
.
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d
C
(
G
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,
a
1
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a
2
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a
2
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···
=
a
q
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-
S
=
{
w
}
,
A
1
=
{
u
1
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}
,
V
(
A
i
)=
{
v
i
}
(
i
=2
,
3
,
···
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−
3),
V
=
{
v
2
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2
,
···
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n
−
3
}
.
d
A
1
ä
k
Ï
f
.
5
Œ
A
1
Ó
u
K
3
…
|
E
(
S,V
(
A
1
)
∪
V
)
|
=
n
−
2.
Ø
”
˜
„
5
,
·
‚
©
wu
1
/
∈
E
(
G
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½
ö
wu
n
−
3
/
∈
E
(
G
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ü
«
œ
¹
?
Ø
.
d
ž
,
wu
2
,wu
3
∈
E
(
G
),
wv
i
∈
E
(
G
)(
i
=2
,
3
,
···
,n
−
4).
X
ã
4,
-
c
(
u
1
u
2
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,c
(
u
1
u
3
)=
2
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(
u
2
u
3
)=3
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(
wu
2
)=4
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(
wu
3
)=5
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(
wv
2
)=6
,c
(
wv
3
)=7.
du
G
(
S
∪
V
(
A
1
))
•
¹
˜
‡
ç
ô
o
,
d
Ú
n
2.2
·
‚
k
E
T
−
n
(
V
)=
∅
.
du
E
(
A
1
,V
)=3
n
−
7
−
(
n
+1)=2
n
−
8,
=
V
¥
z
˜
‡
:Ñ
–
†
A
1
¥
ü
‡
:
ƒ
.
Ø
”
v
2
u
3
,v
3
u
1
∈
E
(
T
−
n
).
wu
1
∈
E
(
G
),
wv
n
−
3
/
∈
E
(
G
)
ž
,
d
T
−
n
Ø
¹
ç
ô
3
K
2
,
¤
±
c
(
v
2
u
3
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∈{
1
,
7
}
,
c
(
v
3
u
1
)
∈{
3
,
6
}
.
d
ž
,
{
v
2
u
3
,v
3
u
1
,wu
2
}
´
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‡
ç
ô
3
k
2
,
†
b
g
ñ
.
wu
1
/
∈
E
(
G
),
wv
n
−
3
∈
E
(
G
)
ž
,
-
c
(
w v
4
)=8.
du
T
−
n
Ø
¹
ç
ô
3
K
2
,
¤
±
c
(
v
2
u
3
)
∈{
1
,
7
}∩{
1
,
8
}
.
Ï
d
,
c
(
v
2
u
3
)=1.
du
T
−
n
Ø
¹
ç
ô
3
K
2
,
¤
±
c
(
v
3
u
1
)=7.
d
ž
,
{
v
2
u
3
,v
3
u
1
,wv
4
}
´
˜
‡
ç
ô
3
K
2
,
†
b
g
ñ
.
e
s
=2,
q
=
n
−
2.
d
ž
|
C
(
G
)
|
=
∅
,
a
1
=
a
2
=
···
=
a
n
−
2
=1.
-
S
=
{
w
1
,w
2
}
,
V
(
A
i
) =
{
v
i
}
(
i
= 1
,
2
,
···
,n
−
2),
V
=
{
v
2
,v
2
,
···
,v
n
−
2
}
.
w
1
w
2
/
∈
E
(
G
)
ž
,
3
V
¥–
k
n
‡
:Ý
•
2.
Ø
”
˜
„
5
,
·
‚
d
G
(
v
1
) =
d
G
(
v
2
) =
d
G
(
v
3
) = 2.
{
ü
©
Û
Œ
G
[
w
1
,w
2
,v
i
,v
j
](1
≤
i,j
≤
3)
•
¹
˜
‡
ç
ô
o
,
d
Ú
n
2.2
Œ
E
T
−
n
(
V
)=
∅
…
E
T
−
n
(
{
v
i
}
,
{
v
4
,v
5
,
···
,v
n
−
2
}
)=
∅
(
i
=1
,
2
,
3).
Ï
d
,
E
T
−
n
≤
2(
n
−
2) + 1+ 3=2
n
≤
3
n
−
7,
=
n
=7.
d
ž
w
1
,w
2
,v
1
,v
2
,v
3
†
K
5
Ó
,
ù
†
T
−
n
´
²
¡
ã
g
ñ
.
w
1
w
2
∈
E
(
G
)
ž
,
X
J
3
{
v
1
,v
2
,
···
,v
n
−
2
}
¥–
k
n
‡
:Ý
•
2.
Ø
”
˜
„
5
,
·
‚
d
G
(
v
1
)=
d
G
(
v
2
)=
d
G
(
v
3
)=2,
d
c
¡
y
²
Œ
±
í
Ñ
g
ñ
.
Ï
d
,
3
{
v
1
,v
2
,
···
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n
−
2
}
¥•
•
3
ü
‡
:Ý
•
2.
Ø
”
˜
„
5
,
·
‚
d
G
(
v
1
)=
d
G
(
v
2
)=2,
d
G
(
v
2
)=
d
G
(
v
3
)=
···
=
d
G
(
v
n
−
2
)=1.
Ø
”
˜
„
5
,
-
{
w
1
v
3
,w
2
v
4
}⊆
E
(
G
).
d u
E
(
{
w
1
,w
2
}
,V
)=3
n
−
7
−
(
n
+1)=2
n
−
8,
=
V
¥
z
‡
:Ñ
†du
v
1
,v
2
ƒ
.
{
ü
©
Û
Œ
c
(
v
2
v
3
)
∈{
c
(
w
1
v
1
)
,c
(
w
2
v
4
)
}
,
c
(
v
1
v
4
)
∈{
c
(
w
1
v
3
)
,c
(
w
2
v
2
)
}
.
d
ž
,
{
w
1
w
2
,v
2
v
3
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1
v
4
}
´
˜
‡
ç
ô
3
K
2
,
†
b
g
ñ
.
d
d
,
·
‚
y
²
n
≥
7,
AR
(
T
−
n
,
3
K
2
)
≤
n
.
½½½
nnn
3.3.
é
?
¿
n
≥
8
,
AR
(
T
−
n
,
4
K
2
) = 2
n
−
2
.
y
²
.
Ä
k
y
²
e
.
.
˜
‡
ã
G
∈
T
−
n
,
§
:
8
•
V
(
G
)=
{
v
1
,v
2
,
···
,v
n
}
,
>
8
•
E
(
G
)=
{
v
1
v
2
}∪{
v
1
v
i
,v
1
v
i
|
3
≤
i
≤
n
}∪{
v
i
v
i
+1
|
3
≤
i
≤
n
−
2
}
.
é
u
ã
G
,
Ä
k
r
¤
k †
v
1
,v
2
ƒ
2
n
−
3
^
>
/þ
Ø
Ó
ô
Ú
.
é
u
ã
G
¥
•
e
>
,
·
‚
^
Ó
˜
«
#
ô
Ú
5
/
.
d
ž
,
ã
G
¥
v
k
ç
ô
4
K
2
d
d
·
‚
,
é
?
¿
n
≥
8,
AR
(
T
−
n
,
4
K
2
)
≥
2
n
−
2.
é
u
þ
.
,
·
‚
^
‡
y
{
.
b
^
2
n
−
1
«
ô
Ú
5
/
T
−
n
>
,
d
ž
T
−
n
¥
Ø
¹
ç
ô
4
K
2
.
w
,
,
T
−
n
¥
k
ç
ô
3
K
2
.
-
G
´
T
−
n
ç
ô
)
¤
f
ã
,
w
,
,
E
(
G
)= 2
n
−
1.
d
ž
,
G
¥
¹
k
ç
ô
3
K
2
Ø
¹
ç
ô
4
K
2
.
d
Ú
n
2.1,
•
3
V
(
G
)
˜
‡
f
8
S
,
¦
o
(
G
−
S
)
−|
S
|
=
n
−
6.
-
|
S
|
=
s
,
o
(
G
−
S
)
−|
S
|
=
q
.
-
G
−
S
Û
©
|
•
A
1
,A
1
,
···
,A
q
Ù
¥
|
A
i
|
=
a
i
(1
≤
i
≤
q
).
Ø
”
˜
„
5
,
·
‚
a
1
≥
a
2
≥···≥
a
q
≥
1.
-
G
−
S
ó
©
|
•
C
(
G
)=
G
−
q
S
i
=1
V
(
A
i
)
∪
S
.
du
q
=
s
+
n
−
6
…
s
+
q
≤
n
,
Ï
d
0
≤
s
≤
3.
DOI:10.12677/aam.2022.111030242
A^
ê
Æ
?
Ð
{
b
§
u
²
e
s
= 0,
q
=
n
−
6.
a
1
≤
5
ž
,
|
E
(
G
)
|≤
3
×
5
−
7 = 11
<
2
n
−
1,
†
b
g
ñ
.
Ï
d
a
1
= 7,
a
2
=
a
3
=
···
=
a
q
= 1.
d
ž
|
E
(
G
)
|≤
3
×
7
−
7 = 14
<
2
n
−
1,
†
b
g
ñ
.
e
s
= 1,
q
=
n
−
5.
a
1
≤
3
ž
,
|
E
(
G
)
|≤
n
−
1+6 =
n
+5
<
2
n
−
1,
†
b
g
ñ
.
Ï
d
a
1
= 5,
a
2
=
a
3
=
···
=
a
q
= 1.
d
ž
|
E
G
(
S
∪
V
(
A
1
))
|≤
3
×
6
−
7 = 11,
|
E
(
G
)
|≤
11+
n
−
6 =
n
+5
<
2
n
−
1,
†
b
g
ñ
.
e
s
=2,
q
=
n
−
4.
d
ž
a
1
≤
3,
a
2
=
a
3
=
···
=
v
q
=1.
-
S
=
{
w
1
,w
2
}
,
A
i
=
{
v
i
}
(2
≤
i
≤
q
).
w
,
,
H
=
G
[
{
w
1
,w
2
,v
2
,
···
,v
n
−
2
}
]
−
E
G
(
{
w
1
,w
2
}
)
´
˜
‡
k
n
−
3
‡
:
Ü
²
¡
ã
.
Ï
d
,
|
E
(
H
)
|≤
2(
n
−
3)
−
4=2
n
−
10,
|
E
G
(
S
∪
C
(
G
)
∪
V
(
A
1
))
|≤
3
×
5
−
7=8.
¤
±
|
E
(
G
)
|≤
8+
|
E
(
H
)
|
= 2
n
−
2
<
2
n
−
1,
†
b
g
ñ
.
e
s
= 3,
q
=
n
−
3,
a
2
=
a
3
=
···
=
v
q
= 1.
-
S
=
{
w
1
,w
2
,w
3
}
,
V
(
A
i
) =
{
v
i
}
(1
≤
i
≤
n
−
3),
V
{
v
1
,v
2
,
···
,v
n
}
.
w
,
,
G
−{
e
|
e
∈
E
G
(
S
)
}
]
´
˜
‡
k
n
‡
:
Ü
²
¡
ã
,
=
|
E
(
G
)
|≤
2
n
−
4+3 =
2
n
−
1.
d
ž
,
G
[
S
]
∼
=
K
3
,
Ø
”
˜
„
5
,
d
G
(
v
1
) =
d
G
(
v
2
) =3,
d
G
(
v
i
) =2(3
≤
i
≤
n
−
3).
w
,
,
v
1
v
2
/
∈
E
(
T
−
n
),
Ä
K
T
−
n
(
S
∪{
v
1
,v
2
}
)
∼
=
K
5
,
ù
†
T
−
n
´
²
¡
ã
g
ñ
.
d
²
¡
ã
Ø
¹
K
3
,
3
Œ
E
(
T
−
n
)
\
E
(
G
)
à:Ñ
3
V
¥
.
·
‚
*
G
[
w
1
,w
2
,w
3
,v
1
,v
2
,v
i
](3
≤
i
≤
n
−
3)
•
¹
˜
‡
ç
ô
8
.
d
Ú
n
2.3
Œ
E
T
−
n
(
{
v
1
v
2
,
···
,v
n
−
3
}
) =
∅
.
¤
±
|
E
T
−
n
(
{
v
1
,v
2
}
,
{
v
3
v
4
,
···
,v
n
−
3
}
)
|
= 3
n
−
7
−
(2
n
−
1) =
n
−
6
≥
2.
-
a
=
|
E
T
−
n
(
{
v
1
}
,
{
v
3
v
4
,
···
,v
n
−
3
}
)
|
,
b
=
|
E
T
−
n
(
{
v
2
}
,
{
v
3
v
4
,
···
,v
n
−
3
}
)
|
.
Ø
”
˜
„
5
,
·
‚
a
≥
b
.
n
≥
9
ž
·
‚
k
a
≥
2.
-
v
1
v
3
∈
T
−
n
,
X
J
c
(
v
2
w
i
)
6
=
c
(
v
1
v
3
)(1
≤
i
≤
3),
@
o
v
1
v
3
Ú
E
G
[
w
1
,w
2
,w
3
,v
2
,v
4
,v
5
]
¤
˜
‡
ç
ô
4
K
2
,
†
b
g
ñ
.
X
J
b
≥
1,
d
ž
•
3
ü
^
>
,
Ø
”
•
v
1
v
3
,v
2
v
4
,
·
‚
k
c
(
v
1
v
3
)
∈{
c
(
v
2
w
1
)
,c
(
v
2
w
2
)
,c
(
v
2
w
3
)
}
,
c
(
v
2
v
4
)
∈{
c
(
v
1
w
1
)
,c
(
v
1
w
2
)
,c
(
v
1
w
3
)
}
.
d
ž
,
v
1
v
3
,v
2
v
4
Ú
E
G
(
{
w
1
,w
2
,w
3
,v
5
}
)
|
¤
˜
‡
ç
ô
4
K
2
,
†
b
g
ñ
.
X
J
b
=0,
d
ž
v
1
v
i
∈
T
−
n
(3
≤
i
≤
n
−
3).
d
²
¡
ã
Ø
¹
K
3
,
3
Œ
,
•
3
ü
‡
:
v
i
v
j
(3
≤
i<j
≤
5),
¦
N
S
(
v
i
)
6
=
N
S
(
v
j
).
Ø
”
i
=3
,j
= 4,
d
ž
G
[
{
v
2
,v
3
,v
4
,w
1
w
2
,w
3
}
]
•
¹
˜
‡
ç
ô
8
.
d
Ú
n
2.3
±
9
v
1
v
5
∈
E
T
−
n
Œ
T
−
n
k
˜
‡
ç
ô
4
K
2
,
†
b
g
ñ
.
n
= 8
ž
,
·
‚
k
a
+
b
= 2.
a
=
b
= 1
ž
†
n
≥
9
œ
¹
ƒ
Ó
.
a
= 2
,b
= 0
ž
,
Ø
”
{
v
1
v
3
,v
1
v
4
}⊂
E
(
T
−
n
).
du
d
G
(
v
4
) =
d
G
(
v
5
) = 2,
Ø
”
˜
„
5
,
·
‚
{
v
4
w
1
,v
4
w
2
,v
5
w
1
,v
5
w
2
}⊂
E
(
G
),
d
ž
{
v
4
w
1
,v
5
w
2
,v
2
w
3
,v
1
v
3
}
´
˜
‡
ç
ô
4
K
2
,
†
b
g
ñ
.
d
d
,
·
‚
y
²
é
?
¿
n
≥
8,
AR
(
T
−
n
,
4
K
2
)
≤
2
n
−
2.
Figure4.
wu
1
∈
E
(
G
)or
wv
n
−
3
∈
E
(
G
)
ã
4.
wu
1
∈
E
(
G
)
½
wv
n
−
3
∈
E
(
G
)
DOI:10.12677/aam.2022.111030243
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{
b
§
u
²
Figure5.
k
=6
,n
=13
ã
5.
k
=6
,n
=13
4.
ç
ô
kK
2
½½½
nnn
4.1.
é
?
¿
k
≥
6
…
n
≥
3
k
−
7
,
2
n
+3
k
−
16
≤
AR
(
T
−
n
,kK
2
)
≤
2
n
+4
k
−
15
.
y
²
.
Ä
k
y
²
e
.
.
·
‚
E
˜
‡
ã
G
∈
T
−
n
§
:
8
•
V
(
G
)=
{
v
1
,v
2
,
···
,v
n
}
.
·
‚
3
k
−
4
´
P
1
=
v
2
v
3
···
v
k
−
2
Ä
:
þ
E
ã
G
.
1
˜
Ú
,
·
‚
3
P
1
Ä
:
þ
V
\
˜
^
>
v
1
v
2
,
-
P
1
þ
z
˜
‡
:Ñ
©
O
†
v
1
Ú
v
2
ƒ
,
Ù
¥
v
1
v
3
v
2
´
¡
.
1
Ú
,
3
v
1
v
i
v
i
+1
,v
2
v
i
v
i
+1
(3
≤
i
≤
k
−
3)
±
9
v
1
v
2
v
k
−
2
S
Ü
©
O
V
\
˜
‡
:
,
V
\
2
k
−
9
:
•
g
•
v
k
−
1
,
···
,v
3
k
−
11
.
-
V
\
:
†
þ
n
‡
:Ñ
ƒ
,
d
ž
·
‚
˜
‡
k
3
k
−
11
‡
:
²
¡
n
¿
©
ã
.
1
n
Ú
,
V
\
˜
^
n
−
3
k
+ 11
´
P
2
=
v
3
k
−
11
v
3
k
−
10
···
v
3
k
−
7
,
-
P
2
þ
z
˜
‡
:Ñ
©
O
†
v
1
Ú
v
2
ƒ
,
Ù
¥
v
1
v
3
v
2
v
3
k
−
7
´
¡
.
d
d
,
·
‚
E
Ñ
ã
G
.
·
‚
3
ã
5
¥
‰
Ñ
k
=6
ä
N
E
•{
.
é
u
ã
G
,
·
‚
r
P
2
´
þ
¤
k
>
/þ
ô
Ú
1,
•
e
¤
k
>
/þ
2
n
−
3
k
−
17
«
Ø
Ó
ô
Ú
.
d
ž
,
ã
G
¥
Ø
¹
ç
ô
kK
2
.
d
d
·
‚
,
é
?
¿
k
≥
6
…
n
≥
3
k
−
7,
AR
(
T
−
n
,kK
2
)
≥
2
n
+3
k
−
16.
é
u
þ
.
,
·
‚
^
‡
y
{
.
b
,
•
3
k
≥
6
…
n
≥
3
k
−
7,
¦
AR
(
T
−
n
,kK
2
)
≥
2
n
+3
k
−
14.
@
o
,
•
3
T
−
n
˜
‡
2
n
+ 3
k
−
14
>
/Ú
,
¦
ã
T
−
n
Ø
¹
ç
ô
kK
2
.
-
G
´
T
−
n
ç
ô
)
¤
f
ã
,
w
,
E
(
G
)=2
n
+ 3
k
−
14.
d
ž
,
G
¥
¹
k
ç
ô
(
k
−
1)
K
2
Ø
¹
ç
ô
kK
2
.
-
M
=
{
u
i
w
i
∈
E
(
G
):1
≤
i
≤
k
−
1
}
´
G
¥
ç
ô
(
k
−
1)
K
2
.
-
V
(
M
)=
{
u
1
,
···
,u
k
−
1
,w
1
,
···
,w
k
−
1
}
,
R
=
V
(
G
)
\
V
(
M
).
é
u
i
∈
[
k
−
1],
·
‚
Ø
”
b
|
N
G
(
u
i
)
∩
R
|≤|
N
G
(
w
i
)
∩
R
|
.
du
ã
G
¥
•
Œ
š
ê
•
k
−
1,
·
‚
Œ
±
G
[
R
]=
∅
…
w
i
ž
v
i
(1
≥
i
≥
k
−
1)
3
R
¥
v
k
ú
:
.
é
?
¿
i
∈
[
k
−
1],
|
N
G
(
u
i
)
∩
R
|
=0
k
|
N
G
(
w
i
)
∩
R
|≥
0,
|
N
G
(
u
i
)
∩
R
|
=1
ž
k
N
G
(
u
i
)
∩
R
=
N
G
(
w
i
)
∩
R
.
·
‚
Ø
”
,
1
≤
i
≤
l
ž
,
k
|
N
G
(
u
i
)
∩
R
|
= 1.
l
+1
≤
i
≤
k
−
1
ž
,
k
|
N
G
(
u
i
)
∩
R
|
= 0.
·
‚
k
|
E
G
(
{
u
1
,
···
,u
l
,w
1
,
···
,w
l
}
,R
)
|
= 2
l
.
-
L
=
{
w
l
+1
,
···
w
t
−
1
}
.
l
=
k
−
1
ž
,
k
|
E
G
(
L,R
)
|
= 0.
l
≤
k
−
2
ž
,
k
E
G
(
L,R
)
≤
2(
n
−
(
k
−
1)
−
l
)
−
4 = 2
n
−
2
k
−
2
l
−
2.
d
|
E
G
(
M
)
|≤
3(2
k
−
2)
−
7 = 6
k
−
13
,
|
E
(
G
)
|
=
|
E
G
(
M
)
|
+
|
E
G
(
{
u
1
,
···
,u
l
,w
1
,
···
,w
l
}
,R
)
|
+
DOI:10.12677/aam.2022.111030244
A^
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Ð
{
b
§
u
²
|
E
G
(
L,R
)
|≤
(6
k
−
12)+2
l
+(2
n
−
2
k
−
2
l
−
2) = 2
n
+4
k
−
15,
ù
†
·
‚
b
g
ñ
.
d
d
·
‚
,
é
?
¿
k
≥
6
…
n
≥
3
k
−
7,
AR
(
T
−
n
,kK
2
)
≤
2
n
+4
k
−
15.
ë
•
©
z
[1]Erd˝os, P.,Simonovits,M.andS´os,V.T.(1973)Anti-RamseyTheorems.
Colloquia Mathematica
SocietatisJ´anosBolyai
,
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[2]Li,X.L. and Xu, Z.X. (2009) The Rainbow Number of Matchings in Regular Bipartite Graphs.
AppliedMathematicsLetters
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https://doi.org/10.1016/j.aml.2009.03.019
[3]Schiermeyer,I. (2004) RainbowNumbers for Matchings and Complete Graphs.
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ematics
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[4]Jendrol
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,S.,Schiermeyer,I.andTu,J.H.(2014)RainbowNumbersforMatchingsinPlane
Triangulations.
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https://doi.org/10.1016/j.disc.2014.05.012
[5]Qin,Z.M.,Lan,Y.X.,Shi,Y.T. andYue,J.(2021)Exact Rainbow Numbersfor Matchingsin
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344
,ArticleID:112301.
https://doi.org/10.1016/j.disc.2021.112301
DOI:10.12677/aam.2022.111030245
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