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AdvancesinAppliedMathematicsA^êÆ?Ð,2021,10(11),4047-4055
PublishedOnlineNovemb er2021inHans.http://www.hanspub.org/journal/aam
https://doi.org/10.12677/aam.2021.1011430
Äuã$ŽÛ܇ ŽXÚêïÄ
444ûûûûûû
1∗
§§§>>>ùùù
1†
§§§uuu°°°
2
§§§ŸŸŸwwwAAA
1
1
#õ“‰ŒÆêƉÆÆ§#õ¿°7à
2
#õŒÆêƆXÚ‰ÆÆ§#õ¿°7à
ÂvFϵ2021c1023F¶¹^Fϵ2021c1113F¶uÙFϵ2021c1129F
Á‡
-G=(V(G),E(G))´kn‡º:Úm^>{üëÏã"˜‡Vf:E(G)→{1,2,···,m}
¡•ãG˜‡Û܇ ŽIÒ§XJéuãG¥?¿ü‡ƒº:uÚv÷vω(u)6=
ω(v)§ùpω(u) =
P
e∈E(u)
f(e)§Ù¥E(u)´†:uƒ'é>8Ü"XJ‰ãG¥?¿˜‡
º:vXôÚω(v)§@oãG?¿˜‡Û܇ ŽIÒѬÑãG˜‡~:XÚ"ãG
Û܇ ŽXÚêχ
la
(G)´ãGÛ܇ ŽIÒ¤Ñ¤kXÚ¥•ôÚê"©
̇ïIJL˜ã$Ž(XµlÇã\˜^]!>F
n
+ {e}Ú˜AÏã(ãP
m
(S
n
)ÚV(
ãP
m
(S
l,q
)¿©ã)ƒãÛ܇ ŽXÚ¯K"
'…c
‡ ŽIÒ§Û܇ ŽIÒ§Û܇ ŽXÚ꧿©
ResearchontheLocalAntimagic
ChromaticNumberBasedonGraph
Operations
DandanLiu
1∗
,HongBian
1†
,HaizhengYu
2
,LinaWei
1
1
SchoolofMathematicalSciences,XinjiangNormalUniversity,UrumqiXinjiang
2
CollegeofMathematicsandSystemSciences,XinjiangUniversity,UrumqiXinjiang
∗1˜Šö"
†ÏÕŠö"
©ÙÚ^:4ûû,>ù,u°,ŸwA.Äuã$ŽÛ܇ ŽXÚêïÄ[J].A^êÆ?Ð,2021,10(11):
4047-4055.DOI:10.12677/aam.2021.1011430
4ûû
Received:Oct.23
rd
,2021;accepted:Nov.13
th
,2021;published:Nov.29
th
,2021
Abstract
LetG=(V(G),E(G))beasimpleconnectedgraphwith|V(G)|=nand|E(G)|=m.A
bijectionf: E(G) →{1,2,...,m}iscalledlocalantimagiclabelingifforanytwoadjacent
verticesuandv,ω(u)6=ω(v),whereω(u)=
P
e∈E(u)
f(e),andE(u)isthesetofedges
incidenttou.Thusanylocalantimagiclabelinginducesapropervertexcoloring
ofG,wherethevertexuisassignedthecolorω(u).Thelocalantimagicchromatic
numberχ
la
(G)istheminimumnumberofcolorstakenoverallcoloringsinducedby
localantimagiclabelingsofG.Inthispaper,westudytheexactvaluesofthelocal
antimagicchromaticnumbersofsomegraphsbasedgraphoperation,suchasF
n
+{e},
whereeisapendantedgeaddingtoF
n
andthesub-dividedgraphsP
m
(S
n
)andP
m
(S
l,q
)
ofsomespecialgraphs.
Keywords
AntimagicLabeling,LocalAntimagicLabeling,LocalAntimagicChromaticNumber,
Sub-Divided
Copyright
c
2021byauthor(s)andHansPublishersInc.
This work is licensed undertheCreative Commons Attribution InternationalLicense(CCBY4.0).
http://creativecommons.org/licenses/by/4.0/
1.Úó
-G=(V(G),E(G)) ´˜‡vká:k•{üÕã.¡˜‡Vf :E(G)→
{1,2,···,|E(G)|}•ãG‡ ŽIÒ,XJf÷véuãG?¿ü‡º:uÚvÑkω(u)6=
ω(v),Ù¥ω(u) =
P
e∈E(u)
f(e),E(u)´†º:uƒ'é>8Ü.˜‡ãG¡•‡ Ž,XJ
ãGk˜‡‡ ŽIÒ.ã‡ ŽIÒ½Â•@´dHartsfieldÚRingel[1]31990cÄgJ
Ñ,¦‚„JÑ/ØK
2
±¤këÏ{üãÑk˜‡‡ ŽIÒ0ߎ,–8ù‡ßŽ
ÿ™)û.
CAc,Arumugam<[2]ÚBensmail<[3]©OÕá/JÑ˜‡'‡ ŽIÒƒéf
½Â:Û܇ ŽIÒ,¿…•ÑJÑ/ØK
2
±¤këÏ{üãÑk˜‡Û܇ ŽI
DOI:10.12677/aam.2021.10114304048A^êÆ?Ð
4ûû
Ò0ߎ.¡˜‡Vf:E(G)→{1,2,···,|E(G)|}´ãG˜‡ Û܇ ŽIÒ,XJéu
ãG?Ûü‡ƒº:uÚvÑkω(u) 6= ω(v),Ù¥ω(u) =
P
e∈E(u)
f(e),E(u)´†:uƒ'
é>8Ü.˜‡ãG¡•Û܇ Ž,XJãGk˜‡Û܇ ŽIÒ.eéãG:vX
ôÚω(v),w,ãG?˜‡Û܇ ŽIÒg,/ÑãG˜‡~:XÚ.ÓžArumugam
<[2]JÑÛ܇ ŽXÚê½Â:ãGÛ܇ ŽXÚê´ÙÛ܇ ŽIÒ¥¤¦^
•ôÚê,P•χ
la
(G),¦‚„‰Ñ´!!lÇã!Óã˜AÏãa Û܇ ŽX
Úê(ƒŠ.
Arumugam<[2]‰Ñ´!ãG†k
2
ã!Óã(n≡0(mod4))Û܇ ŽXÚ
ê‰Œ,±9(ã!lÇã!lÇãí˜^>!‚ã!ãÛ܇ ŽXÚê(ƒŠ.
32018c,Lau[4]<ïÄ†ƒ'éãÛ܇ ŽXÚê,±9˜‡ãí½\þ˜^A
½>Û܇ ŽXÚê;3©z[5]¥,ïÄ˜‘ k•:ãÛ܇ ŽXÚê,'X:•
Zã;3©z[6]¥,‰ÑÜãÛ܇ ŽXÚê;ÓcArumugam[7]<‰Ñ´!
!ãK
n
†˜ãK
m
:coronaÈÛ܇ ŽXÚê;Nazula[8]•3Óc‰Ñüã
Û܇ ŽX Úê• x,'X:º8ã!kü^]!>.32020c,Dafik[9]<‰ÑlÇã
¿©ãÛ܇ ŽXÚê(ƒŠ,±9Óã!(ã¿©ãÛ܇ ŽXÚê‰Œ.
ÄuArumugam!Lau!Dafik<ïÄlÇãÚ(ã3˜ã$ŽeÛ܇ ŽXÚê
(J,©ïÄlÇã\˜^]!>Ú(ã!V(ã¿©ãÛ܇ ŽXÚê.
2.̇(J
˜‡lÇã´?¿ü‡º:ÑTÐk˜‡ú:{üã,P•F
n
,Ù¥n•´F
n
¥
n/‡ê.-lÇãF
n
:8•V(F
n
)={u
i
:1≤i≤n}∪{v
i
:1≤i≤n}∪{x},>8
•E(F
n
) ={u
i
v
i
: 1≤i≤n}∪{xu
i
: 1≤i≤n}∪{xv
i
: 1≤i≤n}.be´lÇãF
n
?¿˜
^>,@oF
n
−{e}KL«F
n
í˜^>e.Arumugam<[2]e¡Ún.
Ún1[2]éuãF
n
−{e},e´lÇãF
n
?¿˜^>,@oχ
la
(F
n
−{e}) = 3.
!•ÄlÇãF
n
\þ?¿˜^]!>eƒÛ܇ ŽXÚê,±e(J.
½n2éuãF
n
+{e},e•\3lÇãF
n
(n≥2)þ?¿˜^]!>,K
3 ≤χ
la
(F
n
+{e}) ≤
(
3,e= xv;
4,Ù§.
y²-ãF
n
+{e}:8•V(F
n
+{e}) = V(F
n
)+{v},>8•E(F
n
+{e}) = E(F
n
)+{e}.
Œ±²wχ
la
(F
n
+{e})≥χ(F
n
+{e})= 3.e5,½Â˜‡Vf:E→{1,2,...,3n+1},
K
f(e) = 3n+1;
f(u
i
v
i
) = i,1 ≤i≤n;
f(xu
i
) = 2n+1−i,1 ≤i≤n;
DOI:10.12677/aam.2021.10114304049A^êÆ?Ð
4ûû
Ú
f(xv
i
) = 3n+1−i,1 ≤i≤n.
Šâ±þIÒ,©±eü«œ/?؈:ŠƒÚ:
œ/1]!>e\3lÇãF
n
¥%:xþ.d?ØŒ•e= xv,džˆº:ŠƒÚ•:
ω(v) = 3n+1;
ω(x) = 4n(n+1)+1;
ω(v
i
) = 3n+1,1 ≤i≤n;
Ú
ω(u
i
) = 2n+1,1 ≤i≤n.
²w/,f´ãF
n
+{e}˜‡Û܇ ŽIÒ,…¦^3«pÉôÚ,Ïdχ
la
(F
n
+{e}) ≤3.
χ
la
(F
n
+{e}) = 3.XF
3
+{e}Û܇ ŽXÚ„ã1.
Figure1.Thelocalantimagiccoloring
ofF
3
+{e}
ã1.F
3
+{e}Û܇ ŽXÚ
œ/2]!>e\3lÇãF
n
?¿˜‡Ý:þ.Ø”˜„5,b]!>†:u
1
đ,
=e= u
1
v,džŒ±º:ŠƒÚ•:
ω(v) = 3n+1;
ω(u
1
) = 5n+2;
DOI:10.12677/aam.2021.10114304050A^êÆ?Ð
4ûû
ω(u
i
) = 2n+1,2 ≤i≤n;
w(v
i
) = 3n+1,1 ≤i≤n;
Ú
ω(x) = n(4n+1),1 ≤i≤n.
²w/,f´ãF
n
+ {e}˜‡Û܇ ŽIÒ…¦^4«pÉôÚ,χ
la
(F
n
+ {e})≤4.
XF
3
+{e}Û܇ ŽXÚ„ã2.
Figure2.ThelocalantimagiccoloringofF
3
+{e}
ã2.F
3
+{e}Û܇ ŽXÚ
Ún2[2]éu?¿˜‡kl‡“fäT,kχ
la
(T) ≥l+1.
Ún3[9]ãSS
n,m
´(ã>¿©ã.éun>2Úm>3?¿ê,ãSS
n,m
Û܇
ŽXÚê•n+1 ≤χ
la
(SS
n,m
) ≤n+2.
dÚn3,©é¦(ã¿©ãP
m
(S
n
)Û܇ ŽXÚê•χ
la
(P
m
(S
n
))=n+1
ã.
½n4ãP
m
(S
n
)´(ãS
n
mg>¿©ã,e÷v±e?˜^‡:
1)m= 1,…2 ≤n≤3;
2)m•óê…2m≤n≤2m+1,Kk
χ
la
(P
m
(S
n
)) = n+1.
y²-ãP
m
(S
n
):8•V(P
m
(S
n
)) = {u
0
}∪{u
ki
: 1 ≤k≤m,1≤i≤n}∪{u
0i
: 1 ≤i≤
n},>8•E(P
m
(S
n
))={u
0
u
1i
:1≤i≤n}∪{u
ki
u
k+1i
:1≤k≤m−1,1≤i≤n}∪{u
mi
u
0i
:
1 ≤i≤n}.ãP
m
(S
n
)º:ê•|V(P
m
(S
n
))|= mn+n+1,>ê•|E(P
m
(S
n
))|= mn+n.½Â
˜‡Vf: E→{1,2,...,|E(P
m
(S
n
))|},:
DOI:10.12677/aam.2021.10114304051A^êÆ?Ð
4ûû
œ/1m=1ž,kχ
la
(P
1
(S
2
))≤3,Úχ
la
(P
1
(S
3
))≤4(§‚Û܇ ŽXÚ„eã
3(a)!(b)),f´˜‡Û܇ ŽIÒ¦2 ≤n≤3ž,kP
1
(S
n
) ≤n+1.
Figure3.(a)thelocalantimagiccoloringofP
1
(S
2
);(b)thelocalantimagic
coloringofP
1
(S
3
)
ã3.(a)ãP
1
(S
2
)Û܇ ŽXÚ¶(b)ãP
1
(S
3
)Û܇ ŽXÚ
œ/2m•óêž:
f(u
0
u
1i
) = i,1 ≤i≤n;
f(u
mi
u
0i
) = mn+i,1 ≤i≤n;
f(u
0i
u
mi
) = mn+n+1−i,1 ≤i≤n;
f(u
ki
u
k+1i
) =



mn+(1−k)n+1−i,k≡1(mod2),1 ≤i≤n;
kn+i,k≡0(mod2),1 ≤i≤n.
dþãIÒOŽŒ•,džˆº:ŠƒÚ•:
ω(u
0
) =
n(n+1)
2
;
ω(u
0i
) = mn+i,1 ≤i≤n;
ω(u
ki
) =



mn+1,k≡1(mod2),1 ≤i≤n;
(m+2)n+1,k≡0(mod2),1 ≤i≤n.
k≡1(mod2)ž,ω(u
ki
)=ω(u
01
)=mn+ 1;ω(u
0
)7½†,‡ω(u
0i
)ƒ,2≤i≤n:
n=2mž,ω(u
0
)= ω(u
0
n
2
)=
n(n+1)
2
;n=2m+1ž,ω(u
0
)= ω(u
0n
)=
(2m+1)(2m+1+1)
2
.
²w/,f´˜‡Û܇ ŽIÒ…¦^n+1«pÉôÚ,Ïdχ
la
(P
m
(S
n
)) ≤n+1.
(ã¿©ãäkn‡“f:Ú˜‡¥%:,dÚn2Œ•,χ
la
(P
m
(S
n
)) ≥n+1.nþ¤ã,
÷vþã?˜^‡ž,kχ
la
(P
m
(S
n
)) = n+1.XP
2
(S
4
)Û܇ ŽXÚ„eã4.
½n5ãP
m
(S
l,q
)´V(ãS
l,q
mg>¿©ã, l≥2Úq≥2•?¿êž, Kl+q+1 ≤
P
m
(S
l,q
) ≤l+q+3.
DOI:10.12677/aam.2021.10114304052A^êÆ?Ð
4ûû
Figure4.ThelocalantimagiccoloringofP
2
(S
4
)
ã4.P
2
(S
4
)Û܇ ŽXÚ
y²-ãP
m
(S
l,q
):8•V(P
m
(S
l,q
)) = {u
1
0
,u
2
0
}∪{u
ki
: 1 ≤i≤l,1 ≤k≤m}∪{u
0i
: 1 ≤
i≤l}∪{u
ki
:l+2≤i≤l+q+1,1≤k≤m}∪{u
0i
:l+2≤i≤l+q+1}∪{u
i
:1≤i≤m},
>8•E(P
m
(S
l,q
))={u
1
0
u
1i
:1≤i≤l}∪{u
ki
u
k+1i
:1≤i≤l,1≤k≤m−1}∪{u
mi
u
0i
:1≤
i≤l}∪{u
1
0
u
1
}∪{u
i
u
i+1
: 1≤i≤m−1}∪{u
m
u
2
0
}∪{u
2
0
u
1i
: l+2≤i≤l+q+1}∪{u
ki
u
k+1i
:
l+ 2≤i≤l+ q+ 1,1≤k≤m−1}∪{u
mi
u
0i
:l+ 2≤i≤l+ q+ 1}.ãP
m
(S
l,q
)º:ê
•|V(P
m
(S
l,q
))|=(m+1)(l+q+1)+ 1,>ê•|E(P
m
(S
l,q
))|=(m+1)(l+q+1).½Â˜‡V
f: E→{1,2,...,|E(P
m
(S
l,q
))|},
f(u
m
u
2
0
) = (m+1)(l+q+1)−l;
f(u
0i
u
mi
) = (m+1)(l+q+1)+1−i,1 ≤i≤l;
f(u
0i
u
mi
) = (m+1)(l+q+1)+1−i,l+2 ≤i≤l+q+1;
f(u
1
0
u
1
) =
(
(m−1)(l+q+1)
2
+l+1,1 ≡m(mod2);
m(l+q+1)+
[1−(m−1)](l+q+1)
2
−l,1 ≡m−1(mod2);
f(u
i
u
i−1
) =
(
(m−k)(l+q+1)
2
+l+1,k≡m(mod2);
m(l+q+1)+
[k−(m−1)](l+q+1)
2
−l,k≡m−1(mod2);
f(u
1
0
u
1i
) =
(
(m−1)(l+q+1)
2
+i,1 ≡m(mod2),1 ≤i≤l;
m(l+q+1)+
[1−(m−1)](l+q+1)
2
+1−i,1 ≡m−1(mod2),1 ≤i≤l;
f(u
ki
u
k−1i
) =
(
(m−k)(l+q+1)
2
+i,k≡m(mod2),1 ≤i≤l;
m(l+q+1)+
[k−(m−1)](l+q+1)
2
+1−i,k≡m−1(mod2),1 ≤i≤l;
f(u
2
0
u
1i
) =
(
(m−1)(l+q+1)
2
+i,k≡m(mod2),l+2 ≤i≤l+q+1;
m(l+q+1)+
[1−(m−1)](l+q+1)
2
+1−i,k≡m−1(mod2),l+2 ≤i≤l+q+1;
DOI:10.12677/aam.2021.10114304053A^êÆ?Ð
4ûû
f(u
ki
u
k−1i
) =



(m−k)(l+q+1)
2
+i,k≡m(mod2),l+2 ≤i≤l+q+1;
m(l+q+1)+
[k−(m−1)](l+q+1)
2
+1−i,k≡m−1(mod2),l+2 ≤i≤l+q+1.
d±þIÒŒ,ˆº:ŠƒÚ•:
ω(u
0i
) = (m+1)(l+q+1)+1−i,1 ≤i≤l;
ω(u
1
0
) =



m(l+1)
2
+l+1
2
,k≡m(mod2);
(m+1)(l+1)
2
+l+1
2
,k≡m−1(mod2);
ω(u
0i
) = (m+1)(l+q+1)+1−i,l+2 ≤i≤l+q+1;
ω(u
2
0
) =



mq
2
+q
2
+(m+1)(l+q+1),k≡m(mod2);
(m+1)q
2
+q
2
+(m+1)(l+q+1),k≡m−1(mod2);
ω(u
ki
) =



(m+1)(l+q+1)+1,k≡m(mod2),1 ≤i≤l;
m(l+q+1)+1,k≡m−1(mod2),1 ≤i≤l;
ω(u
i
) =



(m+1)(l+q+1)+1,k≡m(mod2),1 ≤i≤m;
m(l+q+1)+1,k≡m−1(mod2),1 ≤i≤m;
ω(u
ki
) =



(m+1)(l+q+1)+1,k≡m(mod2),l+2 ≤i≤l+q+1;
m(l+q+1)+1,k≡m−1(mod2),l+2 ≤i≤l+q+1.
i=l+q+1,k≡m−1(mod2) ž,kω(u
0l+q+1
)=m(l+q+1)+1.²w/,f´
ãP
m
(S
l,q
)˜‡Û܇ ŽIÒ…¦^l+q+3«pÉôÚ,P
m
(S
l,q
)≤l+ q+ 3.Šâ
Ún2Œ•l+ q+ 1=χ(P
m
(S
l,q
))≤χ
la
(P
m
(S
l,q
)).nþŒ,ãP
m
(S
l,q
)Û܇ ŽXÚê
•l+q+1 ≤P
m
(S
l,q
) ≤l+q+3.XP
2
(S
3,4
)Û܇ ŽXÚ„ã5.
Figure5.ThelocalantimagiccoloringofP
2
(S
3,4
)
ã5.P
2
(S
3,4
)Û܇ ŽXÚ
DOI:10.12677/aam.2021.10114304054A^êÆ?Ð
4ûû
Ä7‘8
I[g,‰ÆÄ7‘8(11761070,61662079);2021c#õ‘Æg£«g,Ä7éÜ‘
8(2021D01C078);2020c#õ“‰ŒÆ˜6;’!˜6‘§‘8]Ï"
ë•©z
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