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AdvancesinAppliedMathematics
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,2021,10(11),3962-3968
PublishedOnlineNovemb er2021inHans.http://www.hanspub.org/journal/aam
https://doi.org/10.12677/aam.2021.1011421
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TheLocalAntimagicChromaticNumber
oftheJoinGraphs
G
∨
K
2
XueYang
1
∗
,HongBian
1
†
,HaizhengYu
2
1
SchoolofMathematicalSciences,XinjiangNormalUniversity,Urumqi Xinjiang
∗
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[J].
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Ð
,2021,10(11):3962-3968.
DOI:10.12677/aam.2021.1011421
È
2
CollegeofMathematicsandSystemSciences,XinjiangUniversity,Urumqi Xinjiang
Received:Oct.23
rd
,2021;accepted:Nov.13
th
,2021;published:Nov.24
th
,2021
Abstract
Let
G
= (
V,E
)
beaconnectedsimplegraphwith
|
V
|
=
n
and
|
E
|
=
m
.Agraph
G
iscalled
localantimagicif
G
hasalocalantimagiclabeling.Abijection
f
:
E
→{
1
,
2
,
···
,m
}
iscalledlocalantimagiclabelingifforanytwoadjacentvertices
u
and
v
,wehave
ω
(
u
)
6
=
ω
(
v
)
,where
ω
(
u
) =
P
e
∈
E
(
u
)
f
(
e
)
,and
E
(
u
)
isthesetofedgesincidentto
u
.Thus
anylocalantimagiclabelinginducesapropervertexcoloringof
G
,wherethevertex
v
isassignedthecolor
ω
(
v
)
.Thelocalantimagicchromaticnumber,denotedby
χ
la
(
G
)
,
istheminimumnumberofcolorstakenoverallcoloringsinducedbylocalantimagic
labelingof
G
.Let
G
and
H
betwovertexdisjointgraphs.Thejoingraphof
G
and
H
,
denotedby
G
∨
H
,isthegraphwhosevertexsetis
V
(
G
)
∪
V
(
H
)
anditsedgesetequals
E
(
G
)
∪
E
(
H
)
∪{
ab
:
a
∈
V
(
G
)
and
b
∈
V
(
H
)
}
.Inthispaper,wegivetheexactvalueof
thelocalantimagicchromaticnumberofthejoingraph
G
∨
K
2
,when
G
ispaths
P
n
,
cycles
C
n
,thestars
S
n
,thefriendshipgraphs
F
n
,respectively.
Keywords
LocalAntimagicLabeling,LocalAntimagicChromaticNumber,JoinGraphs
Copyright
c
2021byauthor(s)andHansPublishersInc.
This work is licensed undertheCreative Commons Attribution InternationalLicense(CCBY4.0).
http://creativecommons.org/licenses/by/4.0/
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.
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2
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i
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n
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2
+
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ó
ê
…
i
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=
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n
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5
n
−
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2
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i
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n
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.
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2
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5
n
−
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2
−
i
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2
,
i
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Û
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,
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n
−
i
2
,
i
´
ó
êž
.
2
-
f
2
(
xy
) = 3
n
.
·
‚
Œ
±
ω
(
v
i
) =
(
9
n
−
6
2
,
i
´
Û
êž
,
11
n
−
2
2
,
i
´
ó
êž
.
ω
(
x
) =
3
n
2
+6
n
2
,
ω
(
y
) =
5
n
2
+4
n
2
.
n
´
ó
êž
,
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‡
Ž
I
Ò
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2
Ñ
ã
P
n
∨
K
2
˜
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…
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^
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Ú
.
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d
n
´
ó
êž
,
χ
la
(
P
n
∨
K
2
)
≤
4.
DOI:10.12677/aam.2021.10114213965
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DOI:10.12677/aam.2021.10114213967
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[1]Hartsfield,N.andRingel,G.(1994)PearlsinGraphTheory.AcademicPress,Inc.,Boston.
[2]Arumugam,S.,Premalatha,K.,Bacˇa,M.andSemaniˇcov´a-Feˇnovˇc´ıkov´a,A.(2017)LocalAn-
timagicVertexColoringofaGraph.
GraphsandCombinatorics
,
33
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https://doi.org/10.1007/s00373-017-1758-7
[3]Bensmail,J.,Senhaji,M.and Lyngsie,K.S.(2017)On aCombinationof the1-2-3 Conjecture
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[4]Lau,G.C.,Shiu,W.C.andNg,H.K.(2020)AffirmativeSolutionsonLocalAntimagicChro-
maticNumber.
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5
,69-78.
[5]Lau,G.C., Shiu,W.C.andNg,H.K.(2021)OnLocal Antimagic Chromatic NumberofCycle-
RelatedJoinGraphs.
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https://doi.org/10.7151/dmgt.2177
[6]Lau, G.C., Shiu, W.C. and Ng,H.K. (2018)On LocalAntimagicChromatic Numberof Graphs
withCut-Vertices.arXiv:1805.04801v6
DOI:10.12677/aam.2021.10114213968
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