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PureMathematics
n
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,2022,12(8),1392-1398
PublishedOnlineAugust2022inHans.http://www.hanspub.org/journal/pm
https://doi.org/10.12677/pm.2022.128152
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ResearchonPropertiesofStarMatchings
ofGraphs
PeirongLi
1
∗
,HongBian
2
†
,HaizhengYu
2
1
SchoolofMathematicalSciences,XinjiangNormalUniversity,UrumqiXinjiang
2
CollegeofMathematicsandSystemSciences,XinjiangUniversity,UrumqiXinjiang
Received:Jul.24
th
,2022;accepted:Aug.24
th
,2022;published:Aug.31
st
,2022
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[J].
n
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,2022,12(8):1392-1398.
DOI:10.12677/pm.2022.128152
o
´
Abstract
Matchingtheoryingraphisaclassicalproblemingraphtheory,andalsohasvery
widelyapplicationsinreallife.However,thetraditionalmatchingtheorycanonly
solvetheproblemofone-to-onepersonnelallocation,andthetraditionalone-to-one
matchingcannotsolvetheproblemofmulti-persongroupallocationthatdetermines
the groupleader,thus putting forward the conceptofstarmatching.Complete bipar-
titegraph
K
1
,s
iscalledstargraph.Twostargraphs
K
1
,s
withoutnocommonvertex
inthegraph
G
arecalledindependentstargraphs,andwecallthattheset
M
K
1
,s
(
G
)
consistingofmutually independentstargraphs
K
1
,s
isa
K
1
,s
-matching ofthegraph
G
,
simplystarmatching.Themaximumcardinalityof
M
K
1
,s
(
G
)
iscalled
K
1
,s
-matching
numberof
G
,simplystarmatchingnumber.Wecallthatavertex
v
ofsomemaxi-
mumstarmatching
M
K
1
,s
(
G
)
issaturatedby
M
K
1
,s
(
G
)
.Ifthemaximumstarmatching
M
K
1
,s
(
G
)
saturated all of vertices of
G
, then
M
K
1
,s
(
G
)
is called perfect starmatching.In
thispaper,byusingthemethodofclassificationdiscussion,wepresentsomebounds
andnecessaryandsufficientconditionsofcontainingstarmatchingsinsomespecial
graphs;Moreover,wegivethesufficientconditionsofcontainingperfectstarmatch-
ingsinperfectbinarytreesandperfectp-arytrees,andgivetheexactlyvaluesof
perfectstarmatchingnumbersofperfectbinarytreesandperfectp-arytrees.
Keywords
Matching, StarMatching, StarMatchingNumber, MaximumStarMatching, Perfect
StarMatching
Copyright
c
2022byauthor(s)andHansPublishersInc.
This work is licensed undertheCreative Commons Attribution International License (CCBY 4.0).
http://creativecommons.org/licenses/by/4.0/
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K
1
,s
-
(
š
¿
‡
^
‡
.
½
n
2.3
-
T
´
n
‡
º:
p
ä
(
n
≥
p
+1).
K
T
k
K
1
,s
-
(
š
…
=
1
≤
s
≤
p
+1.
y
²
:
d
p
ä
½
Â
Œ
±
•
∆(
T
)=
p
+ 1,
¤
±
X
J
T
k
K
1
,s
-
(
š
,
K
1
≤
s
≤
p
+1;
q
Ï
•
p
ä
z
˜
‡
º:
v
i
(
š
“
f
:
)
–
õ
†
Ù
§
p
+ 1
‡
º:
ƒ
,
¤
±
1
≤
s
≤
p
+1
ž
,
T
k
K
1
,s
-
(
š
.
e
¡
‰
Ñ
ã
K
n
(
š
ê
(
ƒ
Š
.
½
n
2.4
é
u
ã
K
n
,
K
m
K
1
,s
(
K
n
) =
b
n/
(1+
s
)
c
.
y
²
.
d
ã
K
n
½
Â
Œ
±
•
K
n
z
˜
‡
º:
v
i
Ý
þ
•
d
K
n
(
v
i
)=
n
−
1,
¤
±
1
≤
s
≤
n
−
1
ž
K
n
þ
k
K
1
,s
-
š
…
m
K
1
,s
(
K
n
)
Œ
–
•
1.
e
¡
©
•
ü
«
œ
¹
y
²
ã
•
Œ
(
š
ê
;
Ø
”
k
=
n/
(1 +
s
),
k
∈
N
∗
(
N
∗
“
L
g
,
ê
8
K
0)
ž
,
d
(
ã
K
1
,s
½
Â
Œ
•
:
|
V
(
K
1
,s
)
|
=1 +
s
;
Ø
”
b
K
n
•
Œ
(
š
ê
•
m
K
1
,s
(
K
n
),
K
K
n
•
Œ
(
š
¥
•
¹
º:
‡
ê
•
(1+
s
)
m
K
1
,s
(
K
n
);
é
u
z
˜
‡
n
= (1+
s
)
k
,
K
∈
N
∗
ž
Œ
±
n
)
•
:
o
U
3
K
n
¥
é
k
‡
Ø
Ó
…
Ñ
•
¹
1+
s
‡
º:
f
ã
G
i
(
i
= 1
,
2
,...,k
),
q
Ï
•
K
n
(
Ù
¥
n
= 1+
s
)
z
˜
‡
º:
o
†
Ù
{
s
‡
º:
ƒ
,
¤
±
G
i
¥
(
š
ê
•
m
K
1
,s
(
G
i
) = 1,
Ï
d
m
K
1
,s
(
K
n
) =
k
,
=
d
ž
•
Œ
(
š
ê
•
m
K
1
,s
(
K
n
) =
k
.
k
∈
R
∗
\
N
∗
(
R
∗
“
L
¢ê
8
K
0)
ž
,
Ó
n
y
²
.
é
u
z
˜
‡
n
=(1+
s
)
k
Œ
±
n
)
•
:
o
U
3
K
n
¥
é
b
k
c
‡
Ø
ƒ
Ó
…
Ñ
•
¹
1+
s
‡
º:
f
ã
G
i
(
i
=1
,
2
,...,
b
k
c
),
¤
±
m
K
1
,s
(
G
i
)=
b
k
c
,
=
d
ž
•
Œ
(
š
ê
m
K
1
,s
(
K
n
)=
b
k
c
.
n
þ
¤
ã
,
=
Œ
é
u
z
˜
‡
n
=(1+
s
)
k
(
Ù
¥
k
∈
R
∗
),
ã
K
n
•
Œ
(
š
ê
•
m
K
1
,s
(
K
n
) =
b
k
c
,
=
m
K
1
,s
(
K
n
) =
b
n/
(1+
s
)
c
.
e
¡
‰
Ñ
k
½º:
±
9
“
f
:
ê
ä
(
š
ê
þ
!
e
.
,
±
9
T
(
š
ê
þ
.
•
3
¿
‡
^
‡
.
Figure2.
Binarytree
T
ã
2.
ä
T
½
n
2.5
-
T
´
º:
ê
•
n
,
“
f
:
ê
•
n
0
ä
.
K
1
≤
m
K
1
,
3
(
T
)
≤
n
2
−
1(
n>
4);
²
L
º:
·
ü
Œ
÷
v
:
m
K
1
,
3
(
T
) =
n
2
−
1
…
=
n
0
+
n
1
+1
3
≥
n
2
−
1.
y
²
:
Ä
k
y
²
1
≤
m
K
1
,
3
(
T
)
≤
n
2
−
1.
¯¢
þ
,
n>
4
ž
,
w
,
k
m
K
1
,
3
(
T
)
≥
1;
y
^
‡
y
{
y
²
m
K
1
,
3
(
T
)
≤
n
2
−
1.
b
m
K
1
,
3
(
T
)
>n
2
−
1,
d
(
ã
K
1
,
3
½
Â
Œ
±
•
(
ã
K
1
,
3
k
4
‡
º:
,
…
4
‡
º:
¥
∆(
K
1
,
3
) = 3,
¤
±
e
m
K
1
,
3
(
T
)
>n
2
−
1,
K
ä
T
•
Œ
K
1
,
3
-
š
ê
¥
Ý
•
3
º:
‡
ê
Œ
u
n
2
−
1
‡
.
qd
ä
½
Â
Œ
•
ä
•
Œ
Ý
•
3,
…
ä
T
¥
k
ü
ˆ
f
ä
º:
ê
•
n
2
,
Ï
d
•
Œ
Ý
•
3
º:
‡
ê
•
n
2
−
1
†
b
g
ñ
,
=
m
K
1
,
3
(
T
)
≤
n
2
−
1,
DOI:10.12677/pm.2022.1281521395
n
Ø
ê
Æ
o
´
n
þ
Œ
•
1
≤
m
K
1
,
3
(
T
)
≤
n
2
−
1(
n>
4).
e
¡
y
²
²
L
º:
·
ü
Œ
÷
v
:
m
K
1
,
3
(
T
) =
n
2
−
1
…
=
n
0
+
n
1
+1
3
≥
n
2
−
1.
7
‡
5
:
m
K
1
,
3
(
T
) =
n
2
−
1,
K
˜
½
k
4(
n
2
−
1)
≤
n
,
q
Ï
•
n
0
∗
0+
n
1
∗
1+
n
2
∗
2 =
n
−
1,
¤
±
4(
n
2
−
1)
≤
2
n
2
+
n
1
+1,
=
n
2
−
1
≤
n
0
+
n
1
+1
3
.
¿
©
5
:
d
ä
V
g
Œ
•
Ý
•
3
º:
ê
T
•
n
2
−
1,
…
†
Ù
{
n
‡
Ø
Ó
º:
ƒ
.
du
ä´
ë
Ï
Ã
ã
…
•
Œ
Ý
•
3,
¤
±
Œ
±
ò
?
¿
ä
±
e
ã
2
/
ª
?
1
±
›
,
I
Ï
L
º:
·
ü
ü
Ø
n
2
−
1
‡
3
ݺ:
üü
ƒ
å
ü
‡
º:
œ
¹
(
X
ã
1(a)
¥
œ
¹
®
ü
Ø
),
Ï
d
3
®
•
Ý
•
3
º:
ê
T
•
n
2
−
1
œ
¹
e
•
I
ä
¥
1
Ý:
Ú
2
Ý:
ê
ƒ
Ú
Œ
u
3(
n
2
−
1),
…
Ï
L
º:
·
ü
ü
Ø
n
2
−
1
‡
3
ݺ:
üü
ƒ
ë
œ
¹
(
X
ã
1(a)
¥
œ
¹
®
ü
Ø
),
=
Œ
Ñ
m
K
1
,
3
(
T
) =
n
2
−
1.
du
n
2
−
1
≤
n
0
+
n
1
+1
3
,
=
3(
n
2
−
1)
≤
n
0
+
n
1
+1,
ä
¥
1
Ý:
Ú
2
Ý
:
ê
©
O
•
n
0
Ú
n
1
+1,
¤
±
Œ
±
Ñ
m
K
1
,
3
(
T
) =
n
2
−
1.
Figure3.
Binarytree
T
0
1
ã
3.
ä
T
0
1
ã
1(
b
)
¥
n
2
−
1 = 4,
n
0
+
n
1
+1
3
=
7
3
,
=
n
2
−
1
≥
n
0
+
n
1
+1
3
,
w
,
m
K
1
,
3
(
T
2
) = 2;
ã
1(
c
)
¥
n
2
−
1 = 6,
n
0
+
n
1
+1
3
= 3,
=
n
2
−
1
≥
n
0
+
n
1
+1
3
,
w
,
m
K
1
,
3
(
T
2
) = 2;
ã
3
=
•
ã
1(
a
)
²
L
º:
·
ü
¤
/
¤
ã
,
w
,
÷
v
m
K
1
,
3
(
T
0
1
)=n
2
−
1 = 2.
e
¡
½
n
2.6
Ú
í
Ø
2.7
‰
Ñ
{
p
ä
Ú
{
ê
©
O
k
{
K
1
,p
-
(
š
Ú
{
K
1
,
2
-
(
š
¿
©
^
‡
,
¿
3
{
p
ä
Ú
{
ê
©
O
k
{
K
1
,p
-
(
š
Ú
{
K
1
,
2
-
(
š
ž
,
‰
Ñ
{
K
1
,p
-
(
š
Ú
{
K
1
,
2
-
(
š
ê
(
ƒ
Š
.
Figure4.
Perfectp-arytree
T
ã
4.
{
p
ä
T
½
n
2.6
-
T
´
n
‡
º:
{
p
ä
.
K
é
u
?
¿
m
= log
p
[1+(
p
−
1)
n
](
m
∈
N
∗
)
k
:
DOI:10.12677/pm.2022.1281521396
n
Ø
ê
Æ
o
´
(i)
m
= log
p
[1+(
p
−
1)
n
]
•
Û
êž
,
ä
T
¥
v
k
{
K
1
,p
-
š
;
(ii)
m
= log
p
[1+(
p
−
1)
n
]
•
ó
êž
,
ä
T
¥
k
{
K
1
,p
-
š
,
…
m
K
1
,p
(
T
) =
p
m
−
1
p
2
−
1
.
y
²
:
(i)
Ø
”
m
•
n
{
p
ä
(
„
ã
4)
ê
,
K
º:
‡
ê
n
†
ê
m
ƒ
m
k
X
e
'
X
:
n
=
p
0
+
p
1
+
p
2
+
···
+
p
m
−
1
,
=
m
= log
p
[1+(
p
−
1)
n
].
qd
{
p
ä
½
Â
Œ
•
∆(
T
) = 1+
p
,
¤
±
é
u
?
¿˜
ˆ
{
p
ä
5
`
˜
½
k
K
1
,p
-
š
.
Ï
•
n
=
p
0
+
p
1
+
p
2
+
···
+
p
m
−
1
= 1+(
p
1
+1)+(
p
2
−
1)+
···
+(
p
m
−
2
+1)+(
p
m
−
1
−
1),
…
x
,
y
©
O
•
Û
ê
Ú
ó
êž
,
p
x
+1 = (
p
+1)(
p
x
−
1
−
p
x
−
2
+
p
x
−
3
−···
+1),
p
y
−
1 = (
p
+1)(
p
−
1)(
p
y
−
2
+
p
y
−
4
+
···
+1).
¤
±
m
•
Û
êž
,
n
i
≡
n
j
≡
1(
mod
(
p
+1)),
Ù
¥
n
i
=
p
m
i
−
1
p
−
1
,n
j
=
p
m
j
−
1
p
−
1
;
m
i
,m
j
•
?
¿
ü
‡
Ø
Ó
Û
ê
.
q
Ï
•
K
1
,p
-
š
¥
•
¹
º:
ê
•
p
+1,
¤
±
{
K
1
,p
-
š
¥
Ú
º
:
ê
•
n
0
= (
p
+1)
m
K
1
,p
(
T
),
ù
†
n
i
≡
n
j
≡
1(
mod
(
p
+1))
g
ñ
.
=
m
•
Û
êž
,
{
p
ä
Ã
{
K
1
,p
-
š
.
(ii)
Ï
•
n
=
p
0
+
p
1
+
p
2
+
···
+
p
m
−
1
=
p
m
−
1
p
−
1
,
m
•
ó
êž
,
Œ
±
Š
n
=
p
2
k
−
1
p
−
1
(
Ù
¥
m
= 2
k
,
k
= 0
,
1
,
2
,
···
),
q
Ï
•
p
2
k
−
1
p
−
1
= (
p
+1)(
p
0
+
p
2
+
p
4
+
···
+
p
2
k
−
2
),
¤
±
(
p
+1)
|
p
2
k
−
1
p
−
1
=
(
p
+1)
|
n
,
Ï
d•
3
˜
‡
ê
q
¦
n
=
p
m
−
1
p
−
1
=(
p
+1)
q
,
¤
±
Œ
±
•
m
•
ó
êž
n
{
p
ä
˜
½
k
q
‡
Ø
Ó
•
¹
p+1
‡
º:
f
ã
.
q
Ï
•
{
p
ä
Ø
“
f
:
z
‡
º:
v
i
þ
–
†
Ø
Ó
p
‡
º:
ƒ
…
K
1
,p
-
š
¥
•
¹
º:
ê
•
p+1,
¤
±
m
=log
p
[1+ (
p
−
1)
n
]
•
ó
êž
k
{
K
1
,p
-
š
.
n
þ
Œ
•
{
p
ä
T
{
K
1
,p
-
š
ê
m
K
1
,p
(
T
)
†
m
ƒ
m
k
±
e
'
X
:
m
K
1
,p
(
T
) =
p
0
+
p
2
+
p
4
+
···
+
p
m
−
2
=
p
m
−
1
p
2
−
1
.
Figure5.
Perfectbinarytree
T
(wheregreenedgesdenote
byastarmatchingof
T
ã
5.
{
ä
T
(
Ù
¥
É
Ú
>
“
L
T
(
š
)
3
{
p
ä
¥
p
= 2
ž
,
Š
â
½
n
2.6
Œ
±
e
¡
í
Ø
2.7
µ
í
Ø
2.7
-
T
´
n
‡
º:
{
ä
.
K
é
u
?
¿
m
= log
2
(1+
n
)(
m
∈
N
∗
)
k
:
(i)
m
= log
2
(1+
n
)
•
Û
êž
,
ä
T
¥
v
k
{
K
1
,
2
-
š
;
(ii)
m
= log
2
(1+
n
)
•
ó
êž
,
ä
T
¥
k
{
K
1
,
2
-
š
,
…
m
K
1
,
2
(
T
) =
2
m
−
1
3
.
ã
5(
a
)
¥
,
m
=4
•
ó
êž
,
w
,
k
{
K
1
,
2
-
š
;
ã
5(
b
)
¥
m
=3
•
Û
êž
,
w
,
Ã
{
K
1
,
2
-
š
.
DOI:10.12677/pm.2022.1281521397
n
Ø
ê
Æ
o
´
Ä
7
‘
8
I
[
g
,
‰
Æ
Ä
7
‘
8
(No.11761070,61662079);2021
c
#
õ
‘Æ
g
£
«
g
,
Ä
7
é
Ü
‘
8
(2021D01C078);
#
õ
“
‰
Œ
Æ
2020
c
˜
6
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[1]Hell,P.andKirkpatrick,D.G.(1986)PackingbyCompleteBipartiteGraphs.
SIAMJournal
onAlgorithmDiscreteMathematics
,
7
,199-209.https://doi.org/10.1137/0607024
[2]Lin,W. andLam,P.C.B. (2009)Star Matching andDistanceTwo Labelling.
TaiwaneseJour-
nalofMathematics
,
13
,211-224.https://doi.org/10.11650/twjm/1500405279
[3]
o
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ä
.
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è
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š
¯
K
[D]:[
a
¬
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©
].
H
®
:
À
H
Œ
Æ
,2019.
https://doi.org/10.27014/d.cnki.gdnau.2019.000170
[4]
Û
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†
,
4
-
.
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š
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†
(
Ã
Î
Ò
)
.
Ê
.
d
A
Š
[J].
p
A^
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ÆÆ
A
6
,2015,30(3):
333-339.
[5]
„„
,
Û
~
†
.
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u
(
š
ê
ã
U
þ
e
.
[J].
þ
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ó
Œ
ÆÆ
, 2020,42(4):317-319+367.
DOI:10.12677/pm.2022.1281521398
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