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PureMathematics
n
Ø
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Æ
,2021,11(11),1911-1917
PublishedOnlineNovemb er2021inHans.http://www.hanspub.org/journal/pm
https://doi.org/10.12677/pm.2021.1111213
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Mycielskian
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TheTotalDominationChromatic
NumbersofMycielskianGraphs
XueYang
1
∗
,HongBian
1
†
,HaizhengYu
2
,LinaWei
1
1
Scho olofMathematicalSciences,XinjiangNormalUniversity,UrumqiXinjiang
2
CollegeofMathematicsandSystemSciences,XinjiangUniversity,UrumqiXinjiang
Received:Oct.16
th
,2021;accepted:Nov.16
th
,2021;published:Nov.24
th
,2021
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n
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,2021,11(11):1911-1917.
DOI:10.12677/pm.2021.1111213
È
Abstract
Let
G
=(
V,E
)
beasimple,connected,finiteandundirectedgraph.Atotaldomina-
tioncoloringofagraph
G
isapropercoloringof
G
inwhichopenneighbourhoodof
eachvertexcontainsatleastonecolorclassandeachcolorclassisdominatedbyat
leastonevertex.Thetotaldominationchromaticnumberof
G
,denotedby
χ
td
(
G
)
,
istheminimumnumberofcolorsrequiredforatotaldominationcoloringof
G
.In
thispaper,wepresenttheupperandlowerboundsoftotaldominationchromatic
numbersofMycielskiangraphofarbitrarilygraph,andobtainexactlyvaluesofthe
totaldominationchromaticnumbersofMycielskiangraphsofsomespecialgraphs.
Keywords
TotalDominationColoring,TotalDominationChromaticNumber,
MycielskianGraphs
Copyright
c
2021byauthor(s)andHansPublishersInc.
ThisworkislicensedundertheCreativeCommonsAttributionInternationalLicense(CCBY4.0).
http://creativecommons.org/licenses/by/4.0/
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DOI:10.12677/pm.2021.11112131912
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a
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y
.
2
e
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n
2.2
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≥
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ž
,
χ
td
(
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n
)=2
d
n
3
e
.
DOI:10.12677/pm.2021.11112131913
n
Ø
ê
Æ
È
½
n
2.3
n
≥
3
ž
,
χ
td
(
µ
(
P
n
))=
(
2
d
n
3
e
+1
,
n
≡
1(
mod
3)
ž
,
2
d
n
3
e
+2
,
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.
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²
{
v
i
|
1
≤
i
≤
n
}
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n
:
8
,
{
v
i
v
i
+1
|
1
≤
i
≤
n
−
1
}
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´
P
n
>
8
.
d
½
n
2.1
Ú
Ú
n
2.2
Œ
χ
td
(
µ
(
P
n
))
≥
2
d
n
3
e
+1.
©
±
e
ü
«
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/
?
Ø´
Mycielskian
ã
›
›
X
Ú
ê
þ
.
:
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/
1
n
≡
1(mod3)
n
=4
ž
,
χ
td
(
µ
(
P
4
))
≥
2
d
4
3
e
+1=5.
ã
µ
(
P
4
)
›
›
X
Ú
¦
^
5
‡
ô
Ú
(
X
ã
1
¤
«
),
χ
td
(
µ
(
P
4
))=5.
d
Ú
n
2.2
Œ
χ
td
(
µ
(
P
4
))=
χ
td
(
P
4
)+1.
Figure1.
Thetotaldominationcoloringof
µ
(
P
4
)
ã
1.
µ
(
P
4
)
›
›
X
Ú
n
≥
7
ž
,
-
n
=3
t
+1.
d
Ú
n
2.2
•
χ
td
(
P
n
)=2
d
3
t
+1
3
e
=2
t
+2,
f
1
=(
V
1
,V
2
,
···
,V
2
t
−
1
,
V
2
t
,V
2
t
+1
,V
2
t
+2
)
´
ã
P
n
˜
‡
›
›
X
Ú
÷
v
:
v
n
−
3
,v
n
−
2
,v
n
−
1
,v
n
©
O
X
ô
Ú
•
2
t
−
1
,
2
t,
2
t
+
1
,
2
t
+2,
K
g
1
=(
V
1
,V
2
,
···
,V
2
t
−
1
,V
2
t
,V
2
t
+1
,V
2
t
+2
∪
U,X
)
´
ã
µ
(
P
n
)
˜
‡
›
›
X
Ú
…
¦
^
2
t
+3
‡
ô
Ú
,
Ï
d
χ
td
(
µ
(
P
n
))
≤
2
t
+3,
χ
td
(
µ
(
P
n
))
≤
χ
td
(
P
n
)+1.
œ
/
2
n
≡
0
,
2(mod3)
d
½
n
2.1
Œ
•
χ
td
(
µ
(
P
n
))
≤
2
d
n
3
e
+2.
‡
y
χ
td
(
µ
(
P
n
))=2
d
n
3
e
+2,
•
I
y
χ
td
(
µ
(
P
n
))
6
=
2
d
n
3
e
+1.
b
χ
td
(
µ
(
P
n
))=2
d
n
3
e
+1.
´
•
•
3
˜
‡
:
x
i
∈
X
X
#ô
Ú
2
d
n
3
e
+1,
@
o
ã
µ
(
P
n
)
Š
:
u
7
X
,
˜
‡
ô
Ú
l
(1
≤
l
≤
2
d
n
3
e
).
3ù
«
œ
/
e
,
•
3
˜
:
v
k
›
›
?
Û
˜
‡
ô
Ú
a
,
g
ñ
.
χ
td
(
µ
(
P
n
))
≥
2
d
n
3
e
+2,
y
.
2
e
¡
Ú
n
´
Chithra
<
3
©
z
[4]
¥
‰
›
›
X
Ú
ê
.
Ú
n
2.4
[4]
n
≥
5
…
n
6
=7
ž
,
χ
td
(
C
n
)=2
d
n
3
e
.
n
=3
ž
,
χ
td
(
C
3
)=3,
Ù
Mycielskian
ã
µ
(
C
3
)
˜
‡
›
›
X
Ú
„
ã
2(a),
Œ
χ
td
(
µ
(
C
3
))=
4.
n
=4
ž
,
χ
td
(
C
4
)=2,Mycielskian
ã
µ
(
C
4
)
˜
‡
›
›
X
Ú
„
ã
2(b),
K
χ
td
(
µ
(
C
4
))
≤
4.
DOI:10.12677/pm.2021.11112131914
n
Ø
ê
Æ
È
w
,
χ
td
(
µ
(
C
4
))
6
=3,
¯¢
þ
,
χ
td
(
µ
(
C
4
))
6
=3
ž
7
k
˜
‡
:
v
k
›
›
?
˜
ô
Ú
a
,
χ
td
(
µ
(
C
4
))
≥
4,
l
χ
td
(
µ
(
C
4
))
≤
4.
n
≥
5
ž
,
e
¡
½
n
‰
Ñ
ã
µ
(
C
n
)
›
›
X
Ú
ê
.
Figure2.
(a)Thetotaldominationcoloringof
µ
(
C
3
);(b)Thetotal
dominationcoloringof
µ
(
C
4
)
ã
2.
(a)
µ
(
C
3
)
›
›
X
Ú
;(b)
µ
(
C
4
)
›
›
X
Ú
½
n
2.5
n
≥
5
ž
,
χ
td
(
µ
(
C
n
))=
(
2
d
n
3
e
+2
,
n
≡
0(
mod
3)
ž
,
2
d
n
3
e
+1
,
Ù
§
.
y
²
{
v
i
|
1
≤
i
≤
n
}
´
C
n
:
8
,
{
v
i
v
i
+1
|
1
≤
i
≤
n
−
1
}∪{
v
n
v
1
}
´
C
n
>
8
.
d
½
n
2.1
Ú
Ú
n
2.4
•
χ
td
(
µ
(
P
n
))
≥
2
d
n
3
e
+1.
y
•
Ä
Mycielskian
ã
›
›
X
Ú
ê
þ
.
,
©
•
±
e
œ
/
:
œ
/
1
n
≡
1(mod3)
ž
,
-
n
=3
t
+1.
ã
C
n
›
›
X
Ú
f
1
X
e
:
f
1
(
v
i
)=
(
2
l
−
1
,
i
=3
l
−
2
½
i
=3
l
ž
,
2
l,
i
=3
l
−
2
ž
.
Ù
¥
1
≤
l
≤
t
−
1,
2
-
f
1
(
v
n
−
3
)=2
t
−
1
,f
(
v
n
−
2
)=2
t,f
(
v
n
−
1
)=2
t
+1,
Ú
f
(
v
n
)=2
t
+
2.
´
•
f
1
=(
V
1
,V
2
,
···
,V
2
t
−
1
,V
2
t
,V
2
t
+1
,V
2
t
+2
)
´
ã
C
n
˜
‡
›
›
X
Ú
,
K
g
1
=(
V
1
,V
2
∪
U,
···
,V
2
t
−
1
,V
2
t
,V
2
t
+1
,V
2
t
+2
,X
)
´
ã
µ
(
C
n
)
˜
‡
›
›
X
Ú
¦
^
2
t
+3
‡
ô
Ú
,
Ï
d
χ
td
(
µ
(
C
n
))
≤
2
d
n
3
e
+1.
n
≡
1(mod3)
ž
,
χ
td
(
µ
(
C
n
))=2
d
n
3
e
+1.
œ
/
2
n
≡
2(mod3)
ž
,
n
=3
t
+2.
ã
C
n
›
›
X
Ú
f
2
X
e
:
f
2
(
v
i
)=
(
2
l
−
1
,
i
=3
l
−
2
½
i
=3
l
ž
,
2
l,
i
=3
l
−
2
ž
.
Ù
¥
1
≤
l
≤
t
,
2
-
f
1
(
v
n
−
1
)=2
t
+1
,f
(
v
n
)=2
t
+2.
w
,
f
2
=(
V
1
,V
2
,
···
,V
2
t
,V
2
t
+1
,V
2
t
+2
)
´
DOI:10.12677/pm.2021.11112131915
n
Ø
ê
Æ
È
ã
C
n
˜
‡
›
›
X
Ú
,
@
o
g
2
=(
V
1
,V
2
∪
U,
···
,V
2
t
+1
,V
2
t
+2
,X
)
•
´
ã
µ
(
C
n
)
˜
‡
›
›
X
Ú
…
¦
^
2
t
+3
‡
ô
Ú
.
a
q
/
,
n
≡
2(mod3)
ž
,
χ
td
(
µ
(
C
n
))=2
d
n
3
e
+1.
œ
/
3
d
½
n
2.1
Œ
•
χ
td
(
µ
(
C
n
))
≤
2
d
n
3
e
+2.
‡
y
χ
td
(
µ
(
C
n
))=2
d
n
3
e
+2,
•
I
y
χ
td
(
µ
(
C
n
))
6
=
2
d
n
3
e
+1.
^
½
n
2.3
y
²
•{
a
q
Œ
χ
td
(
µ
(
C
n
))=2
d
n
3
e
+1
´
Ø
Œ
U
,
χ
td
(
µ
(
C
n
))=
2
d
n
3
e
+2,
y
.
2
e
5
•
Ä
(
ã
S
n
Mycielskian
ã
›
›
X
Ú
ê
.
(
ã
:
8
Ú
>
8
©
O
P
•
V
(
S
n
)=
{
v
}∪{
v
i
|
1
≤
i
≤
n
}
,
E
(
S
n
)=
{
vv
i
|
1
≤
i
≤
n
}
.
w
,
(
ã
›
›
X
Ú
ê´
2,
e
¡
½
n
‰
Ñ
ã
µ
(
S
n
)
›
›
X
Ú
ê
(
ƒ
Š
.
½
n
2.6
n
≥
2
ž
,
χ
td
(
µ
(
S
n
))=4.
y
²
Ò
(
ã
ó
,
Ø
”
˜
„
5
,
:
v
X
ô
Ú
1,
:
v
i
X
ô
Ú
2,
K
f
=(
V
1
,V
2
)
´
(
ã
˜
‡
›
›
X
Ú
.
d
½
n
2.1
Œ
•
3
≤
χ
td
(
µ
(
S
n
))
≤
4.
•
y
χ
td
(
µ
(
S
n
))=4,
•
I
y
χ
td
(
µ
(
S
n
))
6
=3.
b
χ
td
(
µ
(
S
n
))=3,
©
±
e
œ
/
?
Ø
Ù
›
›
X
Ú
:
œ
/
1
3
ã
µ
(
S
n
)
¥
,
S
n
¤
k
€
:
X
#ô
Ú
3,
Š
:
u
X
ô
Ú
1
½
2.
Š
:
u
X
ô
Ú
1
ž
,
:
v
i
(1
≤
i
≤
n
)
v
k
›
›
ô
Ú
a
;
Š
:
u
X
ô
Ú
2
ž
,
:
v
v
k
›
›
ô
Ú
a
,
g
ñ
.
œ
/
2
:
v
i
€
:
x
i
X
#ô
Ú
2,
:
v
€
:
x
X
ô
Ú
3,
:
u
•
U
X
ô
Ú
1,
:
x
v
k
›
›
?
˜
ô
Ú
a
,
g
ñ
.
œ
/
3
:
v
i
€
:
x
i
X
#ô
Ú
3,
:
v
€
:
x
X
ô
Ú
1,
:
u
•
U
X
ô
Ú
2,
:
x
i
(1
≤
i
≤
n
)
v
k
›
›
?
˜
ô
Ú
a
,
g
ñ
.
Ï
d
χ
td
(
µ
(
S
n
))=3
´
Ø
Œ
U
,
y
.
2
e
¡
•
Ä
ã
µ
(
W
n
)
›
›
X
Ú
ê
,Chithra[4]
‰
Ñ
Ó
ã
›
›
X
Ú
ê
(
ƒ
Š
,
3
d
Ä
:
þ
,
½
n
2.8
‰
Ñ
ã
µ
(
W
n
)
›
›
X
Ú
ê
(
ƒ
Š
.
Ú
n
2.7
[4]
n
≥
3
ž
,
χ
td
(
W
n
)=
(
3
,
n
´
ó
êž
,
4
,
n
´
Û
êž
.
½
n
2.8
n
≥
3
ž
,
χ
td
(
µ
(
W
n
))=
(
4
,
n
´
ó
êž
,
5
,
n
´
Û
êž
.
y
²
Ó
ã
:
8
Ú
>
8
©
O
P
•
V
(
W
n
)=
{
v
}∪{
v
i
|
1
≤
i
≤
n
}
,
E
(
W
n
)=
{
vv
i
|
1
≤
i
≤
n
}∪{
v
i
v
i
+1
|
1
≤
i
≤
n
}∪{
v
n
v
1
}
.
w
,
Ó
ã
~
:
X
Ú
•
´
˜
‡
›
›
X
Ú
.
Ø
”
˜
„
5
,
n
´
ó
êž
,
:
v
X
ô
Ú
1,
:
v
i
(
i
´
Û
ê
)
X
ô
Ú
2,
:
v
i
(
i
´
ó
ê
)
X
ô
Ú
3,
Œ
•
f
1
=(
V
1
,V
2
,V
3
)
´
Ó
ã
W
n
˜
‡
›
›
X
Ú
…
¦
^
•
ô
Ú
ê
,
K
g
1
=(
V
1
,V
2
∪
U,V
3
,X
)
´
ã
µ
(
W
n
)
˜
‡
›
›
X
Ú
¦
^
4
‡
ô
Ú
,
χ
td
(
µ
(
W
n
))
≤
4.
d
½
n
2.1
•
,
n
´
ó
êž
,
χ
td
(
µ
(
W
n
))=4.
a
q
/
,
n
´
Û
êž
,
f
2
=(
V
1
,V
2
,V
3
,V
4
)
´
Ó
ã
W
n
˜
‡
›
›
X
Ú
…
¦
^
•
ô
Ú
ê
.
n
´
Û
êž
,
g
2
=(
V
1
,V
2
∪
U,V
3
,V
4
,X
)
´
ã
µ
(
W
n
)
˜
‡
›
›
X
Ú
¦
^
5
‡
ô
Ú
,
χ
td
(
µ
(
W
n
))
≤
5.
DOI:10.12677/pm.2021.11112131916
n
Ø
ê
Æ
È
Ó
n
´
Û
êž
,
χ
td
(
µ
(
W
n
))=5.
2
,
?
Ø
÷
ã
F
n
Mycielskian
ã
µ
(
F
n
)
›
›
X
Ú
ê
.
½
n
2.9
n
≥
3
ž
,
χ
td
(
µ
(
F
n
))=4.
y
²
ã
F
n
:
8P
•
{
v
}∪{
v
i
|
1
≤
i
≤
n
}
,
>
8P
•
{
vv
i
|
1
≤
i
≤
n
}∪{
v
i
v
i
+1
|
1
≤
i
≤
n
−
1
}
.
Ø
”
˜
„
5
,
:
v
X
ô
Ú
1,
:
v
i
(
i
´
Û
ê
)
X
ô
Ú
2,
:
v
i
(
i
´
ó
ê
)
X
ô
Ú
3,
K
f
=(
V
1
,V
2
,V
3
)
´
÷
ã
F
n
˜
‡
›
›
X
Ú
…
¦
^
•
ô
Ú
ê
.
q
Ï
•
χ
(
F
n
)=3,
χ
td
(
F
n
)=3.
y
•
Ä
÷
ã
Mycielskian
ã
.
g
=(
V
1
,V
2
,V
3
∪
U,X
)
´
ã
µ
(
F
n
)
˜
‡
›
›
X
Ú
¦
^
4
‡
ô
Ú
,
Ï
d
χ
td
(
µ
(
F
n
))
≤
4.
d
½
n
2.1
•
χ
td
(
µ
(
F
n
))=4=
χ
td
(
F
n
)+1.
2
•
•
Ä
l
Ç
ã
Mycielskian
ã
›
›
X
Ú
ê
.
½
n
2.10
n
≥
2
ž
,
χ
td
(
µ
(
F
n
))=4.
y
²
ã
F
n
:
8P
•
{
v
}∪{
v
i
|
1
≤
i
≤
2
n
}
,
>
8P
•
{
vv
i
|
1
≤
i
≤
2
n
}∪{
v
i
v
i
+1
|
1
≤
i
≤
2
n
…
i
´
Û
ê
}
.
ã
F
n
˜
‡
›
›
X
ÚX
e
:
:
v
X
ô
Ú
1,
:
v
i
(
i
´
Û
ê
)
X
ô
Ú
2,
:
v
i
(
i
´
ó
ê
)
X
ô
Ú
3,
K
f
=(
V
1
,V
2
,V
3
)
´
l
Ç
ã
F
n
˜
‡
›
›
X
Ú
…
¦
^
•
ô
Ú
ê
.
q
Ï
•
χ
(
F
n
)=3,
χ
td
(
F
n
)=3.
y
g
=(
V
1
,V
2
,V
3
∪
U,X
)
´
ã
µ
(
F
n
)
˜
‡
›
›
X
Ú
¦
^
4
‡
ô
Ú
,
Ï
d
χ
td
(
µ
(
F
n
))
≤
4.
d
½
n
2.1
•
χ
td
(
µ
(
F
n
))=4=
χ
td
(
F
n
)+1.
2
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[1]Gera,R.,Rasmussen,C.W.andHorton,S.(2006)DominatorColoringsandSafeClique
Partitions.
CongressusNumerantium
,
181
,19-32.
[2]Kazemi,A.P.(2015)TotalDominatorChromaticNumberofaGraph.
TransactionsonCom-
binatorics
,
4
,57-68.
[3]Zhou,Y.andZhao,D.(2019)OnDominationColoringinGraphs.ArXiv:1909.03715
[4]Chithra,K.P.andMayamma,J.(2021)TotalDominationColoringofGraphs.
Journalof
MathematicsandComputerScience
,
11
,442-458.
[5]Mycielski,J.(1995)Surlecoloriagedesgraphs.
ColloquiumMathematicum
,
3
,161-162.
https://doi.org/10.4064/cm-3-2-161-162
DOI:10.12677/pm.2021.11112131917
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