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AdvancesinAppliedMathematicsA^êÆ?Ð,2021,10(11),3962-3968
PublishedOnlineNovemb er2021inHans.http://www.hanspub.org/journal/aam
https://doi.org/10.12677/aam.2021.1011421
G†K
2
éãÛ܇ ŽXÚê
ÈÈÈ
1∗
§§§>>>ùùù
1†
§§§uuu°°°
2
1
#õ“‰ŒÆ§êƉÆÆ§#õ¿°7à
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Á‡
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ŽIÒ"ãGÛ܇ ŽIÒ´˜‡Vf:E→{1,2,···,m}§¦éãG?¿ü‡ƒ
º:uÚvÑkω(u)6=ω(v)§Ù¥ω(u)=
P
e∈E(u)
f(e),E(u)´†:uƒ'é>8Ü"
eéãG:vXôÚω(v)§ ²wÑG?˜ ‡Û܇ ŽIÒÑãG˜‡~:XÚ"
ãGÛ܇ ŽXÚê´ÙÛ܇ ŽIÒ¥¤^•ôÚê§P•χ
la
(G)"‰½ü‡:
ØãGÚH§ãGÚHéã§P•G∨H§´3ãGÚHÄ:þ§2òGz˜‡:
†Hz˜‡:ƒëã"©‰Ñ´P
n
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n
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n
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n
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2
éãÛ܇ ŽXÚê(ƒŠ"
'…c
Û܇ ŽIÒ§Û܇ ŽXÚê§éã
TheLocalAntimagicChromaticNumber
oftheJoinGraphsG∨K
2
XueYang
1∗
,HongBian
1†
,HaizhengYu
2
1
SchoolofMathematicalSciences,XinjiangNormalUniversity,Urumqi Xinjiang
∗1˜Šö"
†ÏÕŠö"
©ÙÚ^:È,>ù,u°.G†K
2
éãÛ܇ ŽXÚê[J].A^êÆ?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
LetG= (V,E) beaconnectedsimplegraphwith|V|= nand|E|= m.AgraphGiscalled
localantimagicifGhasalocalantimagiclabeling.Abijectionf:E→{1,2,···,m}
iscalledlocalantimagiclabelingifforanytwoadjacentverticesuandv,wehave
ω(u) 6= ω(v),whereω(u) =
P
e∈E(u)
f(e),andE(u)isthesetofedgesincidenttou.Thus
anylocalantimagiclabelinginducesapropervertexcoloringofG,wherethevertexv
isassignedthecolorω(v).Thelocalantimagicchromaticnumber,denotedbyχ
la
(G),
istheminimumnumberofcolorstakenoverallcoloringsinducedbylocalantimagic
labelingofG.LetGandHbetwovertexdisjointgraphs.ThejoingraphofGandH,
denotedbyG∨H,isthegraphwhosevertexsetisV(G)∪V(H)anditsedgesetequals
E(G) ∪E(H)∪{ab:a∈V(G)andb∈V(H)}.Inthispaper,wegivetheexactvalueof
thelocalantimagicchromaticnumberofthejoingraphG∨K
2
,whenGispathsP
n
,
cyclesC
n
,thestarsS
n
,thefriendshipgraphsF
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/
1.0
-G=(V,E)´äkn‡:!m^>ëÏ{üã.31990c,HartsfieldÚRingel[1]Äg
JÑã‡ ŽIÒ½Â,¿JÑ/˜‡ãG¡•‡ Ž,XJãGk˜‡‡ ŽIÒ.0¡
˜‡Vf:E(G)→{1,2,···,m}•ãG‡ ŽIÒ,XJf÷véuãG?¿ü‡º:u
ÚvÑkω(u) 6= ω(v),Ù¥ω(u) =
P
e∈E(u)
f(e),E(u)´†º:uƒ'é>8Ü.
2017 c,Arumugam <[2]ÚBensmail <[3]Äu‡ Ž½Â©OÕá/JÑ˜
DOI:10.12677/aam.2021.10114213963A^êÆ?Ð
È
‡'‡ ŽIÒƒéf½Â:Û܇ ŽIÒ.ãGÛ܇ ŽIÒ´˜‡Vf :
E→{1,2,···,m},¦éãG?¿ü‡ƒº:uÚvÑkω(u)6=ω(v),Ù¥ω(u)=
P
e∈E(u)
f(e),E(u)´†:uƒ'é>8Ü.eéãG:vXôÚω(v),w,ãG?˜‡Û
܇ ŽIÒg,/ÑãG˜‡~:XÚ.¡ãG´Û܇ Ž,KGk˜‡Û܇ Ž
IÒ.ÓžArumugam<3©z[2]¥JÑÛ܇ ŽXÚê½Â:ãGÛ܇ ŽXÚê
´ÙÛ܇ ŽIÒ¥¤¦^•ôÚê,P•χ
la
(G).
‰½ü‡:ØãGÚH,ãGÚHéã,P•G∨H,´3ãGÚHÄ:þ,2òG
z˜‡:†Hz˜‡:ƒëã.˜‡lÇã´?¿ü ‡º:ÑTÐk˜‡ú:
{üã,P•F
n
,Ù¥n•´F
n
¥n/‡ê.3©z[2]¥Arumugam<‰Ñ´P
n
!
S
n
!lÇãF
n
!ÜãK
2,n
˜AÏãaÛ܇ ŽXÚê(ƒŠ,„‰Ñ:ê–
•4ãG†K
2
éãÛ܇ ŽXÚêþ!e.,Ù¥K
2
´K
2
Öã.
2018c,Lau<‰ÑeZãaÛ܇ ŽXÚê.3©z[4]¥,¦‚ÄgJÑ©z
[2]¥Arumugam<'uãχ
la
(G∨K
2
)e.˜‡‡~,¿‰ÑA½^‡eãχ
la
(G∨K
n
)
Û܇ ŽXÚê;e..Lau<[4]„)û©z[2]¥Arumugam<JÑ¯K3.3
Ú½n2.15,¦‚„‰Ñ?¿˜‡ÜãK
m,n
Û܇ ŽXÚê(ƒŠ.Óž¦
‚[5]„ïÄ˜†ƒ'éãÛ܇ ŽXÚ.2019c,Lau<[6]Ñ‰½]!>,‘
k•:ãÛ܇ ŽXÚê;e.,¿)û©z[4]ߎ2,Ü©)û©z[2]
¯K3.1.
©ïÄ´P
n
,C
n
,(ãS
n
,±9lÇãF
n
†ãK
2
éãÛ܇ ŽXÚê(
ƒŠ.
2.̇(J
ù!‰ÑãG´´P
n
,C
n
,(ãS
n
,±9lÇãF
n
ž,†K
2
éãÛ܇ ŽX
Úê(ƒŠ.
Ún2.1[2]‰½?˜ãG,Kχ
la
(G) ≥χ(G).
½n2.2n≥2ž,χ
la
(P
n
∨K
2
) = 4.
y.ãP
n
∨K
2
Û܇ ŽXÚêe.w,´4.¯¢þ,K
4
´ãP
n
∨K
2
˜‡:Ñf
ã,ŠâÚn2.1·‚kχ
la
(P
n
∨K
2
) ≥χ(P
n
∨K
2
) = 4.e5¦ãP
n
∨K
2
Û܇ ŽX Úê
þ..
ãP
n
ÚK
2
:8©OP•{v
i
:1≤i≤n}Ú{x,y}.KãP
n
∨K
2
>8´E(P
n
∨K
2
)=
{xv
i
,yv
i
: 1≤i≤n}∪{v
i
v
i+1
: 1≤i≤n−1}∪{xy}.ÏdãP
n
∨K
2
kn+2‡:,3n^>.·
‚én©ü«œ/?Ø:
œ/1.n´Ûê.
DOI:10.12677/aam.2021.10114213964A^êÆ?Ð
È
½ÂVf
1
: E(P
n
∨K
2
) →{1,2,···,3n},éãP
n
∨K
2
>?1XeIÒ:
f
1
(v
i
v
i+1
) =
(
n−
i+1
2
,i´Ûêž,
i
2
,i´óêž.
f
1
(xv
i
) =







3n−3
2
+
i+1
2
,i´Ûê…i6= nž,
n−1+
i
2
,i´óêž,
5n−3
2
,i= nž.
f
1
(yv
i
) =
(
3n−
i+1
2
,i´Ûêž,
5n−3
2
−
i
2
,i´óêž.
2-f
1
(xy) = 3n.·‚Œ±
ω(v
i
) =
(
11n−5
2
,i´Ûêž,
9n−5
2
,i´óêž.
ω(x) =
3n
2
+6n−1
2
,
ω(y) =
5n
2
+4n+1
2
.
n´Ûêž,Û܇ ŽIÒf
1
ÑãP
n
∨K
2
˜‡~:XÚ…¦^4‡ØÓô
Ú.Ïdn´Ûêž,χ
la
(P
n
∨K
2
) ≤4.
œ/2.n´óê.
½ÂVf
2
: E(P
n
∨K
2
) →{1,2,···,3n},‰ãP
n
∨K
2
>?1XeIÒ:
f
2
(v
i
v
i+1
) =
(
n−
i+1
2
,i´Ûêž,
i
2
,i´óêž.
f
2
(xv
i
) =







3n−2
2
+
i
2
,i´óê…i6= nž,
n−1+
i+1
2
,i´Ûêž,
5n−2
2
,i= nž.
f
2
(yv
i
) =
(
5n−2
2
−
i+1
2
,i´Ûêž,
3n−
i
2
,i´óêž.
2-f
2
(xy) = 3n.·‚Œ±
ω(v
i
) =
(
9n−6
2
,i´Ûêž,
11n−2
2
,i´óêž.
ω(x) =
3n
2
+6n
2
,
ω(y) =
5n
2
+4n
2
.
n´óêž,Û܇ ŽIÒf
2
ÑãP
n
∨K
2
˜‡~:XÚ…¦^4‡ØÓô
Ú.Ïdn´óêž,χ
la
(P
n
∨K
2
) ≤4.
DOI:10.12677/aam.2021.10114213965A^êÆ?Ð
È
nþŒ•,χ
la
(P
n
∨K
2
) = 4.
e5•ÄãC
n
∨K
2
(n≥3)Û܇ ŽXÚ.3©z[5]¥,Lau<®²‰Ñn´óê
ž,ãC
n
∨K
2
Û܇ ŽXÚê´4.·‚•Än´Ûêž,ãC
n
∨K
2
Û܇ ŽXÚê.
½n2.3n´Ûêž,χ
la
(C
n
∨K
2
) = 5.
y.n´Ûêž,ãC
n
∨K
2
~:XÚê´5.ŠâÚn2.1χ
la
(C
n
∨K
2
)≥χ(C
n
∨
K
2
) = 5.•y²χ
la
(C
n
∨K
2
) = 5,•Iy²χ
la
(C
n
∨K
2
) ≤5.
ãC
n
∨K
2
:8Ú>8©OP•
V(C
n
∨K
2
) = {v
i
: 1 ≤i≤n}∪{x,y},
E(C
n
∨K
2
) = {xv
i
,yv
i
: 1 ≤i≤n}∪{v
i
v
i+1
: 1 ≤i≤n}.
Ù¥>v
n
v
n+1
´>v
n
v
1
.½ÂVf:E(C
n
∨K
2
)→{1,2,···,3n+1},éãC
n
∨K
2
>IÒX
e:
f(xv
i
) = n+i,1 ≤i≤nž,
f(yv
i
) = 3n+1−i,1 ≤i≤nž,
f(v
i
v
i+1
) =
(
n−
i−1
2
,i´Ûêž,
i
2
,i´óêž.
Ùg,-f(xy)=3n+ 1.ÏLOŽŒ,:v
1
¤XôÚω(v
1
)=
11n+3
2
.i´Ûê…i6=1
ž,:v
i
¤XôÚω(v
i
)=5n+ 1,i´óêž,:v
i
¤XôÚω(v
i
)=5n+ 2;:x¤Xô
Úω(x) =
3n
2
+7n+2
2
;:y¤XôÚω(y) =
5n
2
+7n+2
2
.w,,ù´5‡pÉôÚ,Kf´ãC
n
∨K
2
˜‡Û܇ ŽIÒ…χ
la
(C
n
∨K
2
) ≤5,χ
la
(C
n
∨K
2
) = 5,y.
,•ÄãS
n
∨K
2
(n≥2)Û܇ ŽXÚ.Œ±wÑãS
n
∨K
2
•´ãC
3
∨K
n
.n´Û
ê…n≥3ž,Lau<3©z[5]¥‰Ñχ
la
(C
3
∨K
n
) = 4.e¡‰Ñn´óêž,ãS
n
∨K
2

Û܇ ŽXÚê.
½n2.4n´óêž,χ
la
(S
n
∨K
2
) = 4.
y.(ãS
n
ÚãK
2
:8©OP•{v,v
i
:1≤i≤n},{x,y}.ãS
n
∨K
2
>8
•E(S
n
∨K
2
) ={vv
i
,xv
i
,yv
i
: 1≤i≤n}∪{vx,vy,xy}.´•ãS
n
∨K
2
kn+3‡:,3n+3^
>.
Äk,¦ãS
n
∨K
2
Û܇ ŽXÚêe..duK
4
´ãS
n
∨K
2
˜‡:Ñfã,K
ŠâÚn2.1Œχ
la
(S
n
∨K
2
)≥χ(S
n
∨K
2
)=4.Ùg¦ÙÛ܇ ŽXÚêþ..½ÂV
f: E(S
n
∨K
2
) →{1,2,···,3n+3},éãS
n
∨K
2
>vv
i
,xv
i
,yv
i
?1XeIÒ:
f(vv
i
) = i+1,1 ≤i≤nž,
f(xv
i
) =
(
3n+4
2
+i,1 ≤i≤
n
2
ž,
n+2
2
+i,
n
2
+1 ≤i≤nž.
f(yv
i
) =
(
3n+3−2i,1 ≤i≤
n
2
ž,
4n+4−2i,
n
2
+1 ≤i≤nž.
DOI:10.12677/aam.2021.10114213966A^êÆ?Ð
È
2é•{3^>?1IÒ:f(vx)=1,f(vy)=3n+3 Úf(xy)=
3n+4
2
.3IÒfeƒ
A/ãS
n
∨K
2
z‡:¤XôÚ.1≤i≤nž,:v
i
¤XôÚω(v
i
)=
9n+12
2
;:v¤X
ôÚω(v)=
n
2
+9n+8
2
;:x¤XôÚω(x)=
3n
2
+7n+6
2
;:y¤XôÚω(y)=
5n
2
+14n+10
2
.w,ù
´4‡p؃ÓôÚ,Kf´ãS
n
∨K
2
˜‡Û܇ ŽIÒ…χ
la
(S
n
∨K
2
)≤4,nþŒ
•χ
la
(S
n
∨K
2
) = 4,y.
••ÄãF
n
∨K
2
Û܇ ŽXÚ.
½n2.5n≥2ž,χ
la
(F
n
∨K
2
) = 5.
y.lÇãF
n
ÚãK
2
:8©OP•{v
i
:1≤i≤2n+1}Ú{x,y},Ù¥v
2n+1
´ãF
n
¥%:.KãF
n
∨K
2
>8P•E(F
n
∨K
2
) = {xv
i
,yv
i
: 1 ≤i≤2n+1}∪{v
2n+1
v
i
: 1 ≤i≤
2n}∪{v
i
v
i+1
: 1 ≤i≤2n…i´Ûê}.´•ãF
n
∨K
2
k7n+3^>.½ÂVf: E(F
n
∨K
2
) →
{1,2,···,7n+3},‰ãF
n
∨K
2
>?1XeIÒ:
f(v
i
v
i+1
) =
i+1
2
,1 ≤i≤2n…i´Ûêž,
f(xv
i
) =
(
3n+
i+1
2
,1 ≤i≤2n…i´Ûêž,
5n+
i
2
,1 ≤i≤2n…i´óêž.
f(yv
i
) =
(
5n+1−
i+1
2
,1 ≤i≤2n…i´Ûêž,
7n+1−
i
2
,1 ≤i≤2n…i´óêž.
f(v
2n+1
v
i
) =
(
2n+1−
i+1
2
,1 ≤i≤2n…i´Ûêž,
3n+1−
i
2
,1 ≤i≤2n…i´óêž.
2é•{>IÒ:f(v
2n+1
x)=7n+ 2,f(v
2n+1
y)=7n+3Úf(xy)=7n+ 1.ÏLOŽŒ
•ãF
n
∨K
2
z‡:¤XôÚ.i´Ûê…i6=2n+1ž,:v
i
¤XôÚω(v
i
)=10n+2;i´
óêž,:v
i
¤XôÚω(v
i
)=15n+2;:v
2n+1
¤XôÚω(v
2n+1
)=4n
2
+15n+ 5;:x¤Xô
Úω(x) = 9n
2
+15n+3;:y¤X ôÚω(x)=11n
2
+15n+4.w,f´ãF
n
∨K
2
˜‡Û܇
ŽIÒ…¦^5‡pÉôÚ,χ
la
(F
n
∨K
2
) ≤5.
duK
5
´ãF
n
∨K
2
˜‡:Ñfã,KŠâÚn2.1kχ
la
(F
n
∨K
2
) ≥χ(F
n
∨K
2
) = 5.n
þŒ•χ
la
(F
n
∨K
2
) = 5,y.
±þ(JŒe¡íØ.íØy²I^©z[2]¥½n.
Ún2.6[2]éuÜãG= K
2,n
,Kk
χ
la
(G) =



2,n´óê…n≥4ž.
3,n´Ûê½n= 2ž.
íØ2.7n≥2ž,eχ
la
(G) = χ(G),Kχ
la
(G∨K
2
) = χ
la
(G)+2.
y.déã½ÂŒ•ãK
2
:¤XôÚpÉ, …ØÓuãG:¤XôÚ, Kχ
la
(G∨K
2
) ≥
χ(G∨K
2
)=χ(G)+2.(ÜÚn2.6 NãGÛ܇ ŽIÒ¦ãG∨K
2
k˜‡ÛÜ
‡ ŽIÒ…¦^χ
la
(G)+ 2 ‡ôÚ,Kχ
la
(G∨K
2
)≤χ
la
(G)+ 2.Ïdeχ
la
(G)=χ(G),
DOI:10.12677/aam.2021.10114213967A^êÆ?Ð
È
Kχ
la
(G∨K
2
) = χ
la
(G)+2.
ÒãK
n
ó,χ
la
(K
n
) = χ(K
n
) = n.
ߎ2.8n≥2ž,eχ
la
(G) = χ(G),Kχ
la
(G∨K
n
) = χ
la
(G)+n.
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8(2021D01C078);2022c#õ‘Æg£«g,Ä7¡þ‘8!“c‘8;2020c#õ“‰ŒÆ
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ë•©z
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