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AdvancesinAppliedMathematicsA^êÆ?Ð,2023,12(6),3030-3038
PublishedOnlineJune2023inHans.https://www.hanspub.org/journal/aam
https://doi.org/10.12677/aam.2023.126304
²¡n¿©ã¥šëÏãAnti-Ramseyê
ÛÛÛÁÁÁëëë
∗
,dddjjj
úô“‰ŒÆêƉÆÆ§úô7u
ÂvFϵ2023c528F¶¹^Fϵ2023c623F¶uÙFϵ2023c630F
Á‡
‰½ãG˜‡>/Ú§XJãG?¿ü^>ôÚÑ؃Ó,@oÒ`ãG´çô"ãH
3 ãG¥anti-Ramsey ê´¦>/ÚãG¥Ø•3?ÛçôfãH•ŒôÚê"ã
anti-Ramsey ê8c2•ïÄ,cÙ´š3õ«ãa¥anti-Ramsey ê2•
\ïÄ"GilboaÚRodittyïÄdëÏ©|¤ã3ã¥anti-Ramsey
ê§šë Ïã3²¡ã¥anti-Ramsey êØš(J"Ø©òUY±ù‡••ï
Ä>/Úã¥C
3
∪tP
2
ù‡šëÏã3²¡n¿©ã¥anti-Ramsey ê§é?¿
n≥2t+3,t≥2,2n+3t−9 ≤AR(T
n
,C
3
∪tP
2
) ≤2n+4t−5"
'…c
çôš§Anti-Ramsey꧲¡n¿©
TheAnti-RamseyNumber
ofUnconnectedGraphsin
PlaneTriangulationGraphs
DonglianLuo
∗
,JunqiGu
SchoolofMathematicalSciences,ZhejiangNormalUniversity,JinhuaZhejiang
Received:May28
th
,2023;accepted:Jun.23
rd
,2023;published:Jun.30
th
,2023
∗ÏÕŠö"
©ÙÚ^:ÛÁë,dj.²¡n¿©ã¥šëÏãAnti-Ramseyê[J].A^êÆ?Ð,2023,12(6):
3030-3038.DOI:10.12677/aam.2023.126304
ÛÁë§dj
Abstract
Givenanedge-coloringofagraphG,GissaidtoberainbowifanytwoedgesofG
receivedifferentcolors.GiventwographsGandH,theanti-RamseynumberofHin
Gisdefinedtobethemaximumnumberofcolorsinanedge-coloredgraphGwhich
containsnorainbowcopiesofH.Theanti-Ramseynumbersforgraphs,especially
matchings,havebeenstudiedinseveralgraphclasses.GilboaandRodittyfocused
ontheanti-Ramseynumberofgraphswithsmallcomponents,buttheresultsofthe
anti-Ramsey number ofgraphswithsmallcomponents inplane grapharefew.Inthis
paper,wecontinuetheworkinthisdirectanddeterminetheanti-Ramseynumberof
C
3
∪tP
2
in plane triangulations,thenwe can get 2n+3t−9 ≤AR(T
n
,C
3
∪tP
2
) ≤2n+4t−5
forn≥2t+3,t≥2.
Keywords
RainbowMatching,Anti-RamseyNumber,PlaneTriangulations
Copyright
c
2023byauthor(s)andHansPublishersInc.
This work is licensed undertheCreative Commons Attribution InternationalLicense(CCBY4.0).
http://creativecommons.org/licenses/by/4.0/
1.0
‰½˜‡>/ÚãG,XJãGz˜^>þôÚÑ؃Ó,@o¡ãG•çôã.éu‰
½ãGÚH,>/ÚãG¥•3çôfãH•ôÚê,‰H3ãG¥çôê,PŠ
rb(G,H).Ó /,éu‰½ãGÚH,¦>/ÚãG¥Ø•3?ÛçôfãH•ŒôÚ
ê,‰H3ãG¥anti-Ramseyê,PŠAR(G,H).ex(G,H)L«¦ãG¥Ø•¹?Ûf
ãH•Œ>ê.
Anti-Ramsey ê´dErd˝os <[1] uþ-V70c“ÄgJÑ,¿… y²ã3ã¥
anti-Ramsey ê†ãTur´an ê—ƒƒ'.@ÏïÄöŒÑ• Ä1ã´ãœ¹,
õ«ã3ã¥anti-Ramsey ê˜(Ø,'X:[1–3],´[4],ì[5],š[2,6–8] .
©Ì‡•ÄšëÏãanti-Ramseyê¯K.˜‡•-‡ùa ã´š.Schiermeyer[5]ÚChen,
LiÚTu[6]y²š3ã¥anti-Ramseyê.HaasandYoung[7]y²{š3
DOI:10.12677/aam.2023.1263043031A^êÆ?Ð
ÛÁë§dj
ã¥anti-Ramseyê.dJahanbekamÚWest[9] •Äã¥t‡>Ø{š
anti-Ramsey ê.›˜-V±5,éuanti-RamseyêïÄm©dã=•˜AÏãa,
'X:Üã[10–14],²¡ã[15–22],‡ã[23–26] .Cc5,éšëÏãïÄ̇8¥uš
3²¡ã±9‡ãïÄ,'u²¡n¿©ã¥šanti-Ramsey êïÄ(JXe.-T
n
L«d¤kn-º:²¡n¿©ã¤ãq.ÄkJendrol
0
[17] ‰ÑA½š3²¡
n¿©ã¥anti-Ramseyê°(Š,¿‰Ñš3²¡n¿©ã¥anti-Ramseyê
þe..du±þ½nš3²¡n¿©ã¥anti-Ramsey êþ.Úe.åŒ,ƒ
Qin[21],Jendrol
0
[16]ÚChen[15]3[17]Ä:þÅì š3²¡n¿©ã¥
anti-Ramsey êþ..•ªQin [22] (½š3²¡n¿©ã¥anti-Ramsey ê°(
Š.
Øš,GilboaÚRoditty[27] ±9Bialostocki,Gilboa ÚRoditty[28] NõÙ¦šë
Ïã3ã¥anti-Ramsey ê(Ø.Ù¥Bialostocki,Gilboa ÚRoditty[28] ã
¥¤k •õo^>ãanti-Ramsey ê.Gilboa ÚRoditty[27] •¹ššëÏã3
ã¥anti-Ramseyê(Ø,•)P
3
∪tP
2
,P
4
∪tP
2
ÚC
3
∪tP
2
.•¹ššëÏ3²¡n
¿©ãanti-Ramsey ê„vkƒ'(J,Ïd©3±þïÄÄ:þ,ïÄC
3
∪tP
2
ù‡
ëÏ©|¤ã3²¡n¿©ã¥anti-Ramseyê.
••B,e>‰Ñ©ƒ'½ÂÚÎÒ.3ãG¥,V(G) ÚE(G) ©OL«ãGº:8
Ú>8,V(G)¥ƒ•ãGº:,E(G)ƒ•ãG>.d,|V(G)|Úe(G)©OL«ã
G¥º:êÚ>ê.é?¿u,v∈V(G),e∈E(G) ¦e=uv,K¡º:u†v´,•¡u
†vp•Ø,¿…¡º:u(½v) †>e´'é.éu?¿:v∈V(G),†:vƒ'é>
ê8‰:v3ãGÝ,PŠd(v).阇º:x∈V(G),·‚^N
G
(x)L«G¥†xƒ
º:¤|¤º:8.éuãGÚH,e÷vV(H)⊆V(G) ÚE(H)⊆E(G),K¡H´ãG
˜‡fã,PŠH⊆G.eV(H) = V(G),K¡H•G)¤fã.e3G¥¤kë8ÜV(H)
¥ü‡:>ÑÑy38ÜE(H),K¡H•ãGÑfã,P•G[V(H)].éuV(G) ü‡
؃8ÜSÚT,^e
G
(S,T) L«G¥3SÚTƒm¤k>ê8.
2.²¡n¿©ã¥C
3
∪P
2
Anti-Ramseyê
Äk,éC
3
kXe(J.
½n1.[5]é?¿ên≥4,AR(T
n
,C
3
) =

3n−4
2

−1.
½n2.é?¿ên≥6,AR(T
n
,C
3
∪P
2
) =

3n−4
2

−1.
y²:•y²e., ·‚阇ãT
n
∈T
n
>/Ú¦ãT
n
Ø•¹çôC
3
∪P
2
.Šâ½n1,ã
T
n
•3˜‡Ø•¹çôC
3


3n−4
2

−1

->/Úc.ÏdãT
n
>/Úc•Ø•¹çôC
3
∪P
2
.
lkAR(T
n
,C
3
∪P
2
) ≥

3n−4
2

−1.
e5y²ØªAR(T
n
,C
3
∪P
2
)≤

3n−4
2

−1.·‚•I‡y²?¿ãT
n
∈T
n
?¿

3n−4
2

->/Ú•¹˜‡çôC
3
∪P
2
.‡L5,b•3˜‡ãT
n


3n−4
2

->/ÚØ•¹çô
C
3
∪P
2
.Šâ½n1,N´džãT
n
•¹˜‡çôfãH†C
3
Ó.-V(H) ={v
1
,v
2
,v
3
},
DOI:10.12677/aam.2023.1263043032A^êÆ?Ð
ÛÁë§dj
D=V(T
n
)\V(H).Óž-ãG´ãT
n
˜‡>ê•

3n−4
2

çô)¤fã…•¹fãH.é
z‡º:v∈V(G),^d(v) L«ãG¥º:vÝ.Ïdé?¿º:v,w∈Dkvw/∈G.e5
éº:v∈DŠâd(v)©œ¹?Ø.
œ/1.•3˜‡º:v∈D¦d(v) = 3.
du

3n−4
2

≥8,Ïd•3˜‡º:w∈D¦d(w) ≥1.bwv
3
∈E(G).·‚Œ±
T
n
[v
1
,v
2
,v]∪T
n
[w,v
3
]´˜‡çôC
3
∪P
2
,gñ.
œ/2.z‡º:v∈DÑkd(v) ≤2.
D¥Ýê•2º:‡ê•t.éuD¥?¿ü‡Ýê•2º:vÚw.ØJuy,
e-{v
1
v,v
2
v}⊆E(G).XJwv
3
∈E(G),@oŒ±N´é˜‡çôC
3
∪P
2
.Ïdk
{v
1
w,v
2
w}⊆E(G).lŒ±bD¥?¿Ýê•2º:цv
1
,v
2
ƒ.b3D¥•3
ü‡Ýê•2º:v,w¦vw∈E(T
n
).džãG[v
1
,v
2
,v
3
,v,w]X„ãã1¤«.duã
T
n
Ø•¹çôC
3
∪P
2
,•ÄfãT
n
[v
1
,v
2
,v
3
] ∪T
n
[v,w],fãT
n
[v
2
,v,w] ∪T
n
[v
1
,v
3
],±9fã
T
n
[v
1
,v,w]∪T
n
[v
2
,v
3
],Œ±c(vw) ∈{1,2,3}∩{1,5,7}∩{3,4,6}= ∅,gñ.
Figure1.G[v
1
,v
2
,v
3
,v,w],vw∈E(T
n
)
ã1.vw∈E(T
n
)ãG[v
1
,v
2
,v
3
,v,w]
Ïd3D¥é?¿ü‡Ýê•2º:v,wÑkvw/∈E(T
n
).-¤k3D¥÷vd(v)<2
:|¤º:8•V
1
.džãG−V
1
X„ãã2¤«.ØJuy,3ãG−V
1
¥kt‡o¡.džd
uvw/∈E(T
n
),Ïd3ãG−V
1
¥z‡o¡S–k1‡Ýêu2:.b•3º:x∈V
1
3v
1
v
2
vwv
1
Œ¤o¡S…÷vd(x) =1.Ø”bv
1
x∈E(G).ØJuy,3ãT
n
−v
1
v
2
¥¤k
>ôÚþ•c(v
1
v
2
).ÏdfãT
n
[v
1
,x,w]∪T
n
[v
2
,v
3
]´˜‡çôC
3
∪P
2
,gñ.lŒ•é?¿
3o¡Sº:x∈V
1
Ñkd(x) = 0.
Figure2.G−V
1
ã2.G−V
1
Ïd·‚Œ±e(G) ≤3+2t+(n−3−2t) = n.n= 2kž,ke(G) = b
3n−4
2
c= 3k−2.
Ïd3k−2≤n=2k.lkk≤2,gñ.n=2k+ 1ž,ke(G)=

3n−4
2

=3k.Ïd
DOI:10.12677/aam.2023.1263043033A^êÆ?Ð
ÛÁë§dj
3k≤n=2k+ 1.lŒ±k≤1,gñ.ÏdãT
n
?¿

3n−4
2

->/Ú•¹˜‡çô
C
3
∪P
2
,=AR(T
n
,C
3
∪P
2
) ≤

3n−4
2

−1.
nþ¤ã,½n2y²..
3.²¡n¿©ã¥C
3
∪tP
2
Anti-Ramseyê
½n3.é?¿n≥2t+3,t≥2,AR(T
n
,C
3
∪tP
2
) ≥2n+3t−9.
y²:-P´º:^S•v
1
,v
2
,···,v
t−2
˜^´.H´ÏL‰PO\ü‡ƒëº:x,y¿
…rxÚyü‡:†Pþz‡:ƒ ë²¡n¿©ã,H•>²¡±x,y,v
1
Š•>
..Ïdk|H|=t.-T
H
´ÏL‰Hz‡¡FO\˜‡#º:,òÙ†3F¥¤k
º:ƒë¤²¡n¿©ã.ÏdT
H
´º:ê•t+ (2t−4)=3t−4 n¿©ã.-
w´O\3HL¡˜‡#º:.-T´ÏL‰T
H
•¹x,y,w¡O\n−(3t−4) ‡
º:,Ù¥O\º:•w
1
,w
2
,···,w
n−3t+4
,¦{ww
1
,w
n−3t+4
x,w
n−3t+4
y}⊆E(T) …é¤k
i∈{1,···,n−3t+ 3}òw
i
†T¥x,y,w
i+1
ƒë¤n‡º:²¡n¿©ã.
t= 4 …n= 12žãTEX„ãã3¤«.
Figure3.TheconstructionofT,t=4andn=12
ã3.t=4…n=12žãTE
w,,T∈T
n
.-c´T˜‡>/Ú,Äk‰>ww
1
,ww
2
,···,w
n−3t+3
w
n−3t+4
/Ú•1,,
‰¤kT•e>^†1 ØÓ…p؃ÓôÚ/Ú.džŒ±N´uyTØ•¹çôC
3
∪tP
2
…côÚê´(3n−6)−(n−3t+4)+1 = 2n+3t−9.Ïdy²AR(T
n
,C
3
∪tP
2
) ≥2n+3t−9.
nþ¤ã,½n3y²..
½n4.é?¿ên≥2t+3,t≥2,AR(T
n
,C
3
∪tP
2
) ≤2n+4t−5.
y²:·‚ÏLét?18B5y².t=2 ž,é?¿n≥7,y²ØªAR(T
n
,C
3
∪2P
2
)≤
2n+3.·‚ •Iy²?¿ãT
n
∈T
n
?¿(2n+4)->/ÚÑ•¹˜‡çôC
3
∪2P
2
.‡L5,
·‚b•3˜‡ãT
n
(2n+4)->/ÚØ•¹çôC
3
∪2P
2
.Šâ½n2,Œ±N´ãT
n
DOI:10.12677/aam.2023.1263043034A^êÆ?Ð
ÛÁë§dj
•¹˜‡dH
1
ÚH
2
؃¿|¤çôfãH,Ù¥H
1
=C
3
…H
2
=P
2
.-ãG´ãT
n
˜‡>ê•2n+4 çô)¤fã…•¹fãH.-V(H
1
) ={v
1
,v
2
,v
3
},V(H
2
) ={v
4
,v
5
}…
D= V(T
n
)\V(H).é?¿º:v∈V(G),^d(v)L«3ãG¥:vÝ.w,,ãGØ•¹†ã
C
3
∪2P
2
Ófã.Ïdé?¿º:v,w∈Dkvw/∈E(G).lé?¿v∈Dk0 ≤d(v) ≤5.
e5éº:v∈DŠâd(v)?1©œ¹?Ø.
œ/1.•3˜‡º:v∈D¦d(v) = 5.
dž{vv
1
,vv
2
,vv
3
,vv
4
,vv
5
}⊆E(G).XJ•3º:w∈D\{v}Úi∈{1,···,5}¦wv
i
∈
E(G),@oŒ±N´é˜‡çôC
3
∪2P
2
.ÏdŒ±é?¿º:w∈D\{v}Ñkd(w) = 0.
l2n+4 = e(G) = e(G[V(H)])+e
G
(V(H),D) ≤e(T
5
)+5 = 14.Œ±n≤5,gñ.
œ/2.•3˜‡º:v∈D¦d(v) = 4.
b{vv
1
,vv
2
,vv
3
,vv
5
}⊆E(G).XJ•3º:w∈D\{v}Úi∈{1,···,4}¦wv
i
∈
E(G),@oN´é˜‡çôC
3
∪2P
2
.ÏdŒ±é?¿w∈D\{v}Ñkd(w)≤1.l
2n+4 =e(G) = e(G[V(H)])+e
G
(V(H),D)≤e(T
5
)+(n−6)+4 = n+7.Œ±n≤3,g
ñ.
b{vv
1
,vv
2
,vv
4
,vv
5
}⊆E(G).XJ•3º:w∈D\{v}Úi∈{3,4,5}¦wv
i
∈E(G),
@oN´é˜‡çôC
3
∪2P
2
.XJ•3º:w∈D\{v}¦wv
1
∈E(G),@o·‚Œ±u
yG[v,v
4
,v
5
] ∪G[w,v
1
] ∪G[v
2
,v
3
] ´˜‡çôC
3
∪2P
2
,gñ.XJ•3º:w∈D\{v}¦
wv
2
∈E(G),@oŒ±uyG[v,v
4
,v
5
]∪G[w,v
2
]∪G[v
1
,v
3
] ´˜‡çôC
3
∪2P
2
,gñ.ÏdŒ±
é?¿º:w∈D\{v}Ñkd(w) =0.l2n+4=e(G)=e(G[V(H)])+e
G
(V(H),D)≤
e(T
5
)+4 = 13.Œ±n<4,gñ.
œ/3.•3˜‡º:v∈D¦d(v) = 3.
b{vv
1
,vv
2
,vv
3
}⊆E(G).XJ•3º:w∈D\{v}Úi∈{1,2,3}¦wv
i
∈E(G),@
oN´é˜‡çôC
3
∪2P
2
,gñ.ÏdŒ±é?¿º:w∈D\{v}Ñkd(w)≤2.l
2n+4 = e(G) = e(G[V(H)])+e
G
(V(H),D) ≤e(T
5
)+2(n−6)+3 = 2n,gñ.
b{vv
2
,vv
4
,vv
5
}⊆E(G).XJ•3º:w∈D\{v}Úi∈{1,···,5}¦wv
i
∈E(G),
@oN´é˜‡çôC
3
∪2P
2
.ÏdŒ±é?¿º:w∈D\{v}Ñkd(w)=0.l
2n+4 = e(G) = e(G[V(H)])+e
G
(V(H),D) ≤e(T
5
)+3 = 12.Œ±n≤4,gñ.
b{vv
1
,vv
2
,vv
5
}⊆E(G).XJ•3º:w∈D\{v}Úi∈{3,4}¦wv
i
∈E(G),
@oN´é˜‡çôC
3
∪2P
2
.b•3º:w∈D\{v}¦{wv
1
,wv
2
,wv
5
}⊆E(G).
•ÄfãT
n
[v
1
,v
2
,v
3
] ∪T
n
[v,v
4
] ∪T
n
[w,v
5
],fãT
n
[v,v
4
,v
5
] ∪T
n
[v
2
,v
3
] ∪T
n
[w,v
1
],±9fã
T
n
[v,v
4
,v
5
]∪T
n
[v
1
,v
3
]∪T
n
[w,v
2
], Œ±c(vv
4
) ∈{v
1
v
2
,v
1
v
3
,v
2
v
3
,wv
5
}∩{v
4
v
5
,vv
5
,v
2
v
3
,wv
1
}∩
{v
4
v
5
,vv
5
,v
1
v
3
,wv
2
}= ∅, gñ.Ïdé?¿º:w∈D\{v}Ñkd(w) ≤2.l2n+4 = e(G) =
e(G[V(H)])+e
G
(V(H),D) ≤e(T
5
)+2(n−6)+3 = 2n,gñ.
DOI:10.12677/aam.2023.1263043035A^êÆ?Ð
ÛÁë§dj
œ/4.z‡º:v∈DÑkd(v) ≤2.
dž2n+4=e(G)=e(G[V(H)])+e
G
(V(H),D)≤e(T
5
)+2(n−5) =2n−1,gñ.Ïdé
?¿n≥7,AR(T
n
,C
3
∪2P
2
) ≤2n+3.t= 2 ž½n4 y²..
-t≥3.e5y²é?¿n≥2t+ 3kAR(T
n
,C
3
∪tP
2
)≤2n+4t−5.·‚•I
‡y²?¿ãT
n
∈T
n
?¿(2n+4t−4)->/ÚÑ•¹˜‡çôC
3
∪tP
2
.‡L5,·‚
b•3˜‡ãT
n
(2n+4t−4)->/Ú¦ãT
n
Ø•¹çôC
3
∪tP
2
.Šâ8Bb,
Œ±N´ãT
n
•¹˜‡dH
1
ÚH
2
؃¿|¤çôfãH,Ù¥H
1
=C
3
…
H
2
=(t−1)P
2
.-ãG´ãT
n
˜‡>ê•2n+ 4t−4çô)¤fã…•¹fãH.
w,,ãGØ•¹†C
3
∪tP
2
Ófã.-E(H
2
)={u
i
w
i
∈E(G):i∈{1,···,t−1}},
V(H
2
)={u
1
,···,u
t−1
,w
1
,···,w
t−1
},V(H
1
)={v
1
,v
2
,v
3
},±9R=V(G)\V(H).éz‡
i∈{1,···,t−1},·‚Œ±b|N
G
(u
i
) ∩R|≤|N
G
(w
i
) ∩R|.Ï•ãGØ•¹C
3
∪tP
2
,¤
±ãG[R]vk>.Óžéz‡i∈{1,···,t−1},k|N
G
(u
i
) ∩R|=0½ö|N
G
(u
i
) ∩R|=1
…N
G
(u
i
)∩R=N
G
(w
i
)∩R.·‚Œ±bé¤k0≤i≤l>u
1
w
1
,···,u
l
w
l
÷v
|N
G
(u
i
)∩R|=1,é¤kj∈{l+ 1,···,t−1}k|N
G
(u
j
)∩R|=0,Ù¥0≤l≤t−1.
Ïde
G
({u
1
,···,u
l
,w
1
,···,w
l
},R)=2l.-L={v
1
,v
2
,v
3
,w
l+1
,···,w
t−1
}.l=t−1ž,
e
G
(L,R) = 2(n−2t+2)−4 ≤2n−2t−2l−2.l≤t−2ž,duG[L∪R]−E(G[L])´˜‡º
:ê•n−t−l+1²¡Üã,¤±ke
G
(L,R) ≤2(n−t−l+1)−4 = 2n−2t−2l−2.du
ãG´²¡ã,·‚ke(G[V(H)])≤3(2t+1)−6 =6t−3.Ïde(G)=e(G[V(H)])+e
G
({u
1
,··
·,u
l
,w
1
,···,w
l
},R)+e
G
(L,R) ≤(6t−3)+2l+(2n−2t−2l−2) =2n+4t−5,gñ.ly²
AR(T
n
,C
3
∪tP
2
) ≤2n+4t−5.
nþ¤ã,½n4y²..
4.o(
lGilboaÚRoditty[27]•¹ššëÏã3ã¥anti-Ramseyê(Ø¥¼
éu,¿ÀJ²¡n¿©ã•1ã,ïÄ•¹ššëÏãC
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anti-Ramsey ê,ة̇ÏLE4Š/Ú²¡n¿©ã¥C
3
∪tP
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anti-Ramsey
êe.,X$^‡y{,ÏLïÄã(•ªgñ,ly²þ..ù˜ïÄ••8U
YïÄÙ¦•¹ššëÏã3²¡n¿©ã¥anti-RamseyêJøg´.du²¡ã(
'E,,3é(?1©Ûž¬k˜½JÝ,8̇ïÄ••´I‡Øä Ø©Ø
ƒÓþ.Úe.,±9éÙ¦•¹ššëÏã3²¡n¿©ãanti-Ramseyê?1ïÄ.
ë•©z
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