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PureMathematicsnØêÆ,2021,11(8),1585-1600
PublishedOnlineAugust2021inHans.http://www.hanspub.org/journal/pm
https://doi.org/10.12677/pm.2021.118176
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…=ãG´˜‡Ãã"éu,AÏœ¹§XJãG´˜‡Ø¹3-²¡ã§va(G)≤2"
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VertexArboricityProblemofPlanar
GraphswithoutAdjacentShortCycles
XingLiu
∗
,HuijuanWang
†
∗1˜Šö"
†ÏÕŠö"
©ÙÚ^:4(,¦ï.عƒá²¡ã:ÎݯK[J].nØêÆ,2021,11(8):1585-1600.
DOI:10.12677/pm.2021.118176
4(§¦ï
CollegeofMathematicsandStatistics,QingdaoUniversity,QingdaoShandong
Received:Jul.20
th
,2021;accepted:Aug.20
th
,2021;published:Aug.27
th
,2021
Abstract
Intheresearchofsocialnetworks,socialstructuredecompositionhasalwaysbeena
valuableresearchtopic.Forsecurityreasons,westudyanewsocialstructuredecom-
positionproblem,whichcanbedecomposedintoaproblemofminimizationingraph
theory.ThevertexarboricityofagraphG,denotedbyva(G),istheminimumnumber
ofsubsetssuchthattheverticesofGcanbecoloredandeverycolorclassinducesan
acyclicgraphsuchasaforestofG.Normally,va(G) ≤3foreveryplanargraphGand
va(G) ≥1foreverynonemptygraphG.Thereisnodoubtthatva(G) = 1ifandonlyif
Gisanacyclicgraph.Forsomespecialcases,itisknownthatva(G) ≤2ifGisaplanar
graphwithout3-cycles.Recently,RaspaudandWangetal.provedthatva(G)≤2if
Gisaplanargraphwithout4-cyclesorwithout5-cycles.Inaddition,Huang,Shiu,
andWangshowedthatifGisaplanargraphwithout7-cycles,thenva(G) ≤2.Inthis
paper,weprovethatifGisaplanargraphwithoutadjacent3-cyclesand5-cycles,or
withoutadjacent4-cyclesand5-cycles,thenva(G) ≤2.
Keywords
PlanarGraph,VertexArboricity,Adjacent,Cycle
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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DOI:10.12677/pm.2021.1181761588nØêÆ
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DOI:10.12677/pm.2021.1181761589nØêÆ
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a)τ(v→f
3
) =
1
5
§τ(v→f
4
) =
2
5
§XJf´˜‡AÏ4-¡"
b)τ(v→f
3
) =
3
10
§τ(v→f
4
) =
3
10
§XJv
4
´˜5
+
-º:½v
4
´˜AÏ4-º:"
DOI:10.12677/pm.2021.1181761590nØêÆ
4(§¦ï
6)τ(v→f
1
)=
3
5
§τ(v→f
3
)=
2
5
§XJd(f
1
)=d(f
3
)=5§f
3
(v)=2§ü‡3-¡Øƒ§
f
5
(v) = 2§f
6
+
(v) = 1§d(f
4
) = 6
+
§f
1
†f
3
؃"
7)τ(v→f
3
)=
2
5
§τ(v→f
4
)=
2
5
§XJd(f
3
)=d(f
4
)=5§f
3
(v)=2§ü‡3-¡Øƒ§
f
5
(v) = 2,f
6
+
(v) = 1,d(f
1
) = 6
+
§f
3
†f
4
ƒ"
8)τ(v→f) =
3
5
§XJd(f) = 5,f
3
(v) = 2§ü‡3-¡Øƒ§f
5
(v) = 1"
9)τ(v→f) =
3
5
§XJd(f) = 5…f
3
(v) ≤1"
10)τ(v→f) =
1
6
§XJd(f) = 6
+
"
R
v
3.v´†fƒ6
+
-º:"
1)τ(v→f) =
7
5
§XJd(f) = 3"
2)τ(v→f) =
1
2
§XJd(f) = 4"
3)τ(v→f) =
3
5
§XJd(f) = 5"
4)τ(v→f) =
1
6
§XJd(f) = 6
+
"
R
f
1.b½f
1
´(4,4,4)-¡§f
3
(f
1
)=0§d(f)=3§f†f
1
3v:éá§f
1
¥Øv±ü‡º:Ø
´AÏ4-º:§f¥Øv±ü‡º:´5
+
-º:§½öü‡ Ñ´AÏ4-º:§½ö˜‡
´AÏ4-º:,˜‡´5
+
-º:"
1)τ(f→f
1
) =
1
10
§XJf
3
(f) = 1§f¥Øv±ü‡º:ƒ˜´5-º:§ …†n‡3-¡ƒ
§,˜º:•5
+
-º:"
2)τ(f→f
1
) =
3
5
§Ù¦"
éu3-¡f( v
1
,v
2
,v
3
)"^f
1
§f
2
Úf
3
5L«©O†fäkú>v
1
v
2
§v
2
v
3
Úv
3
v
1
¡"
3 -¡f´Ð§XJ÷vµ−3 +τ(v
1
→f) + τ(v
2
→f) + τ(v
3
→f)≥0½−3 +τ(v
1
→
f) + τ(v
2
→f) + τ(v
3
→f) +
1
10
≥0(f÷vR
f
1(1)ž)½−3 + τ(v
1
→f) + τ(v
2
→
f)+τ(v
3
→f)+
3
5
≥0(f÷vR
f
1(2)ž)½−3+τ(v
1
→f)+τ(v
2
→f)+τ(v
3
→f)+τ(f
i
→
f)≥0(i∈1,2,3)½−3+τ(v
1
→f) +τ(v
2
→f) +τ(v
3
→f) +τ(f
i
→f) +τ(f
j
→f)≥0
(i6=j§ij∈1,2,3)"^n
3
g
(f)5L«†fƒÐ3¡ê8"^n
0
f
L«²\º:DŠ
foŠê"
R
f
2.τ(f→f
1
) =
n
0
f
f
3
(f)−n
3
g
(f)
§XJf´˜‡6
+
-¡§…f†f
1
ƒ§f
1
Ø´˜‡Ð3-¡"
R
f
3.f´5¡§f
3
(f)−n
3
g
(f) >0§n
0
f
>0
1)τ(f→f
0
) = n
0
f
§XJf
3
(f)−n
3
g
(f) = 1§…f†f
0
ƒ"f
0
Ø´˜‡Ð3-¡"
2)τ(f→f
0
) =
n
0
f
f
3
(f)−n
3
g
(f)
§XJf
3
(f)−n
3
g
(f) ≥2§…f†f
0
ƒ§f
0
Ø´˜‡Ð3-¡"
ÏL±þDŠ5K§·‚íÑe¡Ún"
Ún4.1†5
+
-º:ƒk
+
-¡–õkk−2(k≥6)‡ØÐ3-¡"
y²µf´ãG¥k-¡(k≥6)"w,n
5
+
(f) = 1"éuk-¡fk≥6§^v
i
(i∈{1,2,···k})L
«÷f^ž\k‡º:"^f
i
(i∈{1,2,···k})L«†fkú>v
i
v
i+1
§^f
k
L«†fkú
>v
k
v
1
"-v
1
´5
+
-º:"e¡•Än«ŒU5µ
DOI:10.12677/pm.2021.1181761591nØêÆ
4(§¦ï
œ/1.bf
4
+
(f) ≥2"w,§f–õ†k−2‡Ø´Ð3-¡ƒ"
œ/2.bf
4
+
(f) = 1"阇†fƒÐ3-¡"XJf
1
´˜4
+
-¡§KÏLR
v
2(2)ÚR
v
3(1)
µτ(v
1
→f
k
) =
7
5
"f
k
´˜‡Ð3-¡"Ó/,XJf
k
´˜‡4
+
-¡§Kf
1
´˜‡Ð3-¡"X
Jf
i
´4
+
-¡§(i∈{2,3,···,k−1})§KÏLR
v
2(1)-(2)ÚR
v
3(1) Œf
1
Úf
k
¥k˜‡´Ð3-¡"
œ/3.bf
4
+
(f) =0"éü‡†fƒ3-¡"XJv
1
´6
+
-º:§KÏLR
v
3(1)-τ(v
1
→
f
1
) =
7
5
Úτ(v
1
→f
k
) =
7
5
"w,§f
1
Úf
k
´Ð3-¡"XJv
1
´5-º:§K?رeü«œ¹µ
•f
3
(v
1
) = 2"KÏLR
v
2(2)-τ(v
1
→f
1
) =
7
5
Úτ(v
1
→f
k
) =
7
5
"f
1
Úf
k
Ñ´Ð3¡"
•f
3
(v
1
) = 3"Ø”˜„5§ÏLR
v
2(1) -τ(v
1
→f
k
) =
7
5
Úτ(v
1
→f
1
) =
17
15
"w,§f
k
´Ð
3-¡§f
6
+
(f
1
) =2§f
3
(f
1
) =1"†f
1
ƒ6
+
-¡Øf±DŠ‰f
1
–•
1
4
"Ïd§ÏL†f
1
ƒ
:DŠÚØf±6
+
-¡DŠf
1
´Ð"
nþ§†5
+
-º:ƒk
+
-¡–õkk−2(k≥6)‡ØÐ3-¡"
Ún4.2XJd(f) = 5§Kf
3
(f)−n
3
g
(f) ≤5n
0
f
"
y²µ-d(f)=5"éu5-¡f,·‚^v
1
§v
2
§v
3
§v
4
Úv
5
L«^ž†fƒº:"f
i
L«†fkú>v
i
v
i+1
(i∈{1,2,3,4})§f
5
L«†fkú>v
5
v
1
"dÚn2.3•§XJf
i
´3-
¡(i∈{1,2,3,4,5})§Kf
3
(f
i
)=0"w,§n
5
+
(f)≥3ž§5n
0
f
≥1≥f
3
(f) −n
3
g
(f)"b
n
4
(f)=5"KÏLR
v
1(6)n
0
f
=0"f
3
(f)>0ž§ŠâÚn2.2 §†fƒ3-¡¥
k˜‡5
+
-º:"Ïd§ÏLR
v
1(2)§R
v
1(4)§R
v
2(2)ÚR
v
3(1)§XJf
i
´3-¡(i∈{1,2,3,4,5})§
Kf
i
´Ð"Ïdf
3
(f)−n
3
g
(f) = 0"e5•Ä±eü«œ/µ
œ/1.bn
5
+
(f) = 1"Ø”˜„5§-v
1
•†fƒ5
+
-º:"•Äo«ŒUµ
(1.1)f
5
+
(f) ≥3"f
3
(f)−n
3
g
(f) = 0§n
0
f
≥0"
(1.2)f
5
+
(f)=2"dé¡5§·‚••Ä8«œ/"w,§f
3
(f) −n
3
g
(f)=0"d(f
1
)=
d(f
i
) ≥5 ½d(f
1
) = d(f
i
) ≥5§i∈{3,4}"Óžn
0
f
≥0"Ïd§·‚•Ä±eü«ŒUµ
1)d(f
1
)=d(f
5
)≥5"ÏLR
v
2(9)ÚR
v
3(3)τ(v
1
→f)=
3
5
§f
3
(f)−n
3
g
(f)≤1"Ïd
f
3
(f)−n
3
g
(f) ≤1 <5n
0
f
"
2)d(f
1
)=d(f
2
)≥5"ÏLR
v
2(5)-(9)ÚR
v
3(3)-τ(v
1
→f)≥
1
5
"Ïdn
0
f
≥0"w,§
f
3
Úf
5
Ñ´Ð3-¡"XJf
4
´˜‡Ð3-¡§Kf
3
(f)−n
3
g
(f) = 0 ≤n
0
f
"y3·‚•Iy²f
4
´
Ð"-f
i
´(v
i
,w
i
,v
i+1
)-¡(i∈{3,4})§f
5
´(v
5
,w
3
,v
1
)-¡"-s
i
5LǠf3v
i
(i∈{3,4,5}):
?éá¡"bw
3
,w
4
Úw
5
¥–k˜‡5
+
-º:"ÏLR
v
2(2)ÚR
f
1(2)f
4
´Ð"b
w
3
,w
4
Úw
5
Ñ´4-º:"Kn
5
+
(s
4
)≥1§n
5
+
(s
5
)≥1"s
1
–õ†(d(s
1
)−3)‡ØÐ3-¡ƒ§
s
2
–õ†(d(s
2
)−3)‡ØÐ3-¡ƒ"é²w§w
3
§w
4
Úw
5
¥–˜‡´AÏžf
4
´Ð"Ï
d·‚•Äw
1
§w
2
Úw
3
ÑØ´AÏ"-h
1
LǠf
4
3w
4
:éá¡"•Ä±eü«ŒU:
•XJd(s
4
) = d(s
5
) ≥6§KÏLR
v
1(1)-(3)ÚR
f
1-2 −3+τ(v
4
→f
4
)+τ(v
5
→f
4
)+τ(w
4
→
f
4
)+τ(h
1
→f
4
)≥
4
5
+
4
5
+
5
6
+
3
5
≥0½−3+ τ(v
4
→f
4
)+τ(v
5
→f
4
)+τ(w
4
→f
4
)+τ(s
4
→
f
4
)+τ(s
5
→f
4
) ≥
4
5
+
4
5
+
5
6
+
2
3
≥0"Ïd§f
4
´Ð"
DOI:10.12677/pm.2021.1181761592nØêÆ
4(§¦ï
•Øw
4
,s
4
Ús
5
–k˜‡´5-¡§XJh
1
¥kü‡5
+
-º:½kü‡AÏ4-º:½k˜A
Ï4-º:Ú˜5
+
-º:§ÏLR
f
3f
4
´Ð"y3·‚=•Äh
1
¥–õk˜‡5
+
-º:½–
õk˜‡AÏ4-º:"XJs
4
Ús
5
¥˜‡´5-¡§,˜‡´6
+
-¡,KÏLR
v
1(1)-(2)ÚR
f
2-3
−3+τ(v
4
→f
4
)+τ(v
5
→f
4
)+τ(w
4
→f
4
)+τ(s
4
→f
4
)+τ(s
5
→f
4
) ≥
4
5
+
4
5
+
4
5
+
1
5
+
1
2
>0"
XJd(s
4
)=d(s
5
)=5§f
3
(s
4
) −n
3
g
(s
4
)≤2žf
3
(s
5
) −n
3
g
(s
5
)≤1½f
3
(s
5
) −n
3
g
(s
5
)≤2ž
f
3
(s
4
)−n
3
g
(s
4
)≤1"Ïd§ÏLR
v
1(1)-(2)ÚR
f
3 −3+ τ(v
4
→f
4
)+ τ(v
5
→f
4
)+ τ(w
4
→
f
4
)+τ(s
4
→f
4
)+τ(s
5
→f
4
) ≥0"
(1.3)f
5
+
(f) = 1.dé¡5§·‚=•Än«œ/µd(f
1
) ≥5§d(f
2
) ≥5Úd(f
3
) ≥5"
1)bd(f
2
) ≥5½d(f
3
) ≥5"ÏLR
v
2(5)-(9)ÚR
v
3(3)τ(v
1
→f) ≥
3
5
§=µ5n
0
f
≥1"
ÏLR
v
1(4)§R
v
2(2)ÚR
v
3(1)f
3
(f)−n
3
g
(f) ≤1"Ïdf
3
(f)−n
3
g
(f) ≤1 ≤5n
0
f
"
2) bd(f
1
) ≥5"ÏLR
v
2(5)-(9)ÚR
v
3(3) τ(v
1
→f) ≥
2
5
§=µ5n
0
f
≥1"w,f
2
Úf
5
Ñ
´Ð3-¡"XJf
3
Úf
4
–k˜‡Ð3-¡§Kf
3
(f) −n
3
g
(f)≤n
0
f
"y3·‚y²f
3
Úf
4
–
˜‡´Ð3-¡"-f
i
´(v
i
,w
i
,v
i+1
)-¡(i∈{2,3,4})§f
5
´(v
5
,w
5
,v
1
)-¡"s
i
LǠf3v
i
(i∈
{2,3,4,5}):?éá¡"bw
2
§w
3
§w
4
Úw
5
–˜‡´5
+
-º:§ÏLR
v
2(2)ÚR
f
1(2)§
f
3
Úf
4
–k˜‡´Ð3-¡"bw
2
§w
3
§w
4
Úw
5
Ñ´4-º:§Ks
3
§s
4
Ús
5
¥Ñ–k˜
‡5
+
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2
§w
3
§w
4
Úw
5
¥–k˜‡´AÏž§f
3
Úf
4
–˜‡´Ð"y3•Äw
2
§
w
3
§w
4
Úw
5
ÑØ´AÏ"h
i
LǠf
i
3w
i
(i∈{3,4}):?éá¡"s
3
§s
4
Ús
5
–ü‡
´5-¡§Øw
i
(i∈{3,4})§XJh
i
¥kü‡5
+
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Ú˜‡5
+
-º:,Kf
i
´Ð"Ïd§XJs
1
§s
2
Ús
3
¥–ü‡´5-¡"K••Äh
3
Úh
4
¥–õ˜
‡5
+
-º:½–õ˜‡AÏ4-º:§k±e8«ŒUµ
XJd(s
3
)=d(s
4
)=5§s
5
≥6§KÏLR
f
2-3-τ(s
3
→f
3
)≥
2
5
§τ(s
4
→f
3
)≥
1
5
½
τ(s
5
→f
4
)≥
1
2
§τ(s
4
→f
4
)≥
1
5
"Ó/§XJd(s
4
)=d(s
5
)=5§s
3
≥6§KÏLR
f
2-3-
τ(s
5
→f
4
)≥
2
5
§τ(s
4
→f
4
)≥
1
5
½τ(s
3
→f
3
)≥
1
2
§τ(s
4
→f
3
)≥
1
5
"XJd(s
3
)=d(s
5
)=5§
s
4
≥6§KÏLR
f
2-τ(s
4
→f
3
)≥
1
2
§τ(s
4
→f
4
)≥
1
2
§ÏLR
f
3-τ(s
3
→f
3
)≥
1
5
§
τ(s
5
→f
4
)≥
1
5
"Ïd§ÏLR
v
1(2)ÚR
f
2–3−3 +τ(v
3
→f
3
) +τ(v
4
→f
3
) +τ(w
3
→
f
3
) + τ(s
3
→f
3
) + τ(s
4
→f
3
)≥0§−3+ τ(v
4
→f
4
) + τ(v
5
→f
4
) + τ(w
4
→f
4
) + τ(s
4
→
f
4
)+τ(s
5
→f
4
)≥0§=µf
3
Úf
4
Ñ´Ð"XJd(s
3
)=d(s
4
)=d(s
5
)=5§Ks
3
Ús
4
¥–˜
‡©ODŠ‰f
3
Úf
4
–
2
5
"KÏLR
v
1(2)ÚR
f
3−3 +τ(v
3
→f
3
) +τ(v
4
→f
3
) +τ(w
3
→
f
3
) + τ(s
3
→f
3
) + τ(s
4
→f
3
)≥0½−3 + τ(v
4
→f
4
) + τ(v
5
→f
4
) + τ(w
4
→f
4
) + τ(s
4
→
f
4
) +τ(s
5
→f
4
)≥0"XJd(s
3
)=d(s
5
)≥6§s
4
=5§Øw
i
(i∈{3,4})§h
i
¥ü‡5
+
-
º:½ü‡AÏ4-º:½˜‡AÏ4-º:Ú˜‡5
+
-º:ž§Kf
i
´Ð"h
3
Úh
4
¥–õ˜
‡5
+
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DOI:10.12677/pm.2021.1181761593nØêÆ
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DOI:10.12677/pm.2021.1181761594nØêÆ
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DOI:10.12677/pm.2021.1181761595nØêÆ
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1
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4
)+ τ(s
2
→f
4
)≥0§=µf
4
´Ð"XJd(s
1
)=d(s
2
)≥6§ÏL
R
f
2kτ(s
1
→f
4
)≥
1
3
§τ(s
2
→f
4
)≥
1
3
"ÏLR
v
1(1)-(3)ÚR
f
2§k−3+τ(v
4
→f
4
)+τ(v
5
→
f
4
)+τ(w
2
→f
4
)+τ(s
1
→f
4
)+τ(s
2
→f
4
) ≥0§=µf
4
´Ð"
2)bf
5
+
(f)=1"dé¡5kn«œ/µd(f
2
)≥5¶d(f
3
)≥5¶d(f
4
)≥5"XJd(f
3
)≥5
½d(f
4
)≥5§ÏLR
v
1(1)-(4)§R
v
2(2)ÚR
v
3(1)f
3
(f)−n
3
g
(f)=0"Ïd§f
3
(f)−n
3
g
(f)=
0 ≤5n
0
f
"XJd(f
2
) ≥5§KÏLR
v
1(1)-(4)§R
v
2(2)§R
v
2(4)-(9)ÚR
v
3(1)§f
3
(f)−n
3
g
(f) ≤
1§τ(v
1
→f)=
3
5
§τ(v
2
→f)≥
2
5
"3)bf
5
+
(f)=0"ÏLR
v
1(1)-(4)§R
v
2(2)§R
v
2(4)–
(9)ÚR
v
3(1)§kf
3
(f)−n
3
g
(f) ≤1§τ(v
1
→f) =
3
5
§τ(v
2
→f) =
3
5
"Ïd§f
3
(f)−n
3
g
(f) ≤5n
0
f
"
nþ¤ã§XJf´G5-¡§Kf
3
(f)−n
3
g
(f) ≤5n
0
f
"
y3·‚y²Gz‡¡fÑkšKŠ"-d(f)≥6"ÏLR
v
1(7)§R
v
2(10)§R
v
3(4)
ÚR
f
2§kc
0
(f)≥0"-d(f)=5§dR
v
1(6)§R
v
2(4)–(9)§R
v
3(3)ÚÚn2.3•§d:D‰5¡
Š–•
1
5
§…f
3
(f)−n
3
g
(f) ≤5n
0
f
"Ïd§c
0
(f) ≥−1+5×
1
5
≥0"-f´G4-¡§KÏL
R
v
1(5)§R
v
2(3)ÚR
v
3(2)§kc
0
(f) = −2+
1
2
+
1
2
+
1
2
+
1
2
= 0"
-d(f) = 3"éu3-¡f§^v
1
§v
2
Úv
3
^žL«fn‡º:"^f
1
§f
2
Úf
3
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ú>v
1
v
2
§v
2
v
3
Úv
3
v
1
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œ/1.n
4
(f)=3"Xã2¤«§s
1
†f3v
1
:?éá§s
2
†f3v
2
:?éá§s
3
†f3v
3
:?
éá"ŠâÚn2.3§f
3
(f) ≤1"f
1
§f
2
Úf
3
¥Ñ–k˜‡5
+
-º:"
(1.1)f
3
(f)=0"bf–†˜‡4
+
-¡éá"KÏLR
v
1(1)-(4)ÚR
f
2-3kc
0
(f)≥−3 +
min{
4
5
+
4
5
+
7
5
,
4
5
+
4
5
+
7
6
+
1
4
,
5
6
+
7
6
+
7
6
,
4
5
+
7
5
+
7
6
,
7
6
+
7
6
+
7
6
}≥0bf†n‡3-¡éá§=µs
1
§
s
2
Ús
3
Ñ´Ð3-¡"dÚn2.3•f
3
(f) = 0§f
4
(f) = 0"?رen«ŒUµ
•bf
5
(f)≥2"w,§f
3
(s
1
)=0§f
3
(s
2
)=0§f
3
(s
3
)=0"XJs
1
§s
2
Ús
3
¥–˜‡
k ü‡5
+
-º:§½ö˜‡5
+
-º:Ú˜‡AÏ4-º:§½öü‡AÏ4-º:§KÏLR
v
1(2)§
R
f
1(2)ÚR
f
3kc
0
(f)≥−3+
4
5
+
4
5
+
4
5
+
3
5
≥0"ÄK§f
1
§f
2
Úf
3
–kü‡I‡‰fDŠ"Ø
”˜„5§-d(f
1
)=d(f
2
)=5§d(f
3
)≥5"ŠâÚn3.1ÚÚn3.2§5
+
¡‰ƒ€3-
DOI:10.12677/pm.2021.1181761596nØêÆ
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¡DŠ–•
1
5
"XJ−3 +
4
5
+
4
5
+
4
5
+ τ(f
2
→f) + τ(f
3
→f)≥0§Kféuf
1
´Ð¶XJ
−3+
4
5
+
4
5
+
4
5
+τ(f
2
→f)+τ(f
3
→f) <0§Kféuf
1
´€"ŠâÚn3.2kτ(f
1
→f) ≥
1
5
"
Ïd§−3+
4
5
+
4
5
+
4
5
+τ(f
2
→f)+τ(f
3
→f)+τ(f
1
→f) ≥0"
•bf
5
(f) = 1§f
6
+
(f) = 2"Ø”˜„5§-d(f
1
) = d(f
3
) ≥6§d(f
2
) = 5"®•f
3
(s
1
) ≤1"
XJf
3
(s
1
) = 1§KÏLR
v
1(1)-(2)ÚR
f
1-3 c
0
(f) ≥−3+
5
6
+
4
5
+
4
5
+min{
3
5
,
1
4
+
1
4
+
1
10
,
1
4
+
1
4
+
1
5
,
1
4
+
1
3
}≥0"XJf
3
(s
1
) = 0§s
1
§s
2
Ús
3
¥–˜‡kü‡5
+
-º:§½ö˜‡5
+
-º:Ú˜‡
AÏ4-º:§½öü‡AÏ4-º:ž§ÏLR
v
1(2)§R
f
1(2)ÚR
f
3 kc
0
(f) ≥−3+
4
5
+
4
5
+
4
5
+
3
5
≥0"
s
1
§s
2
Ús
3
¥k–õ˜‡5
+
-º:½–õ˜‡AÏ4-º:ž§f
i
(i∈{1,2,3})I‡‰fDŠ"Ï
d§ÏLÚn3.1§Ún3.2§R
v
1(1)-(2)ÚR
f
2-3§−3+
4
5
+
4
5
+
4
5
+τ(f
1
→f)+τ(f
3
→f) ≥0
½ö−3+
4
5
+
4
5
+
4
5
+τ(f
2
→f)+τ(f
3
→f)+τ(f
1
→f) ≥0"
•bf
6
+
(f) = 3"ÏLÚn3.1ÚR
f
1τ(f
i
→f) ≥
1
4
i∈{1,2,3}"ÏLR
v
1(1)ÚR
f
2
c
0
(f) ≥−3+
5
6
+
5
6
+
5
6
+
1
4
+
1
4
+
1
4
≥0"
(1.2)f
3
(f)=1"Xã2(b)¤«§bd(f
3
)=3§d(s
1
)=d(s
3
)=d(f
1
)=d(f
2
)≥6"X
Jd(s
2
) = 3§ÏLR
v
1(1)ÚR
f
2kc
0
(f) ≥−3+
5
6
+
5
6
+
5
6
+
1
4
+
1
4
≥0"XJd(s
2
) = 4§ÏLR
v
1(1)§
R
v
1(3)ÚR
f
2kc
0
(f)≥−3+
5
6
+
5
6
+
7
6
+
1
4
≥0"XJd(s
2
)≥5§ÏLR
v
1(1)ÚR
v
1(4)kc
0
(f)≥
−3+
5
6
+
5
6
+
7
5
≥0"
œ/2.n
4
(f) = 2 §n
5
+
(f) = 1"Ø”˜„5§-d(v
3
) ≥5"bf
3
(f) = 0§ŠâR
v
2ÚR
v
3§
÷v^‡5
+
-º:‰3-¡DŠ•
7
5
"Ïd§ÏLR
v
1(1)-(4)§R
v
2(1)-(2)ÚR
v
3(1)kc
0
(f)≥−3 +
4
5
+
4
5
+
7
5
≥0"bf
3
(f)=1§-f
3
´3-¡§Xã2(c)¤«§ŠâÚn2.3§d(f
1
)=d(f
2
)≥6"
XJd(v
3
) = 5§v
3
†n‡3-¡ƒ§ÏLR
v
1(1)§R
v
1(3)-(4)§R
v
2(1)ÚR
f
2c
0
(f) ≥−3+
5
6
+
17
15
+
5
6
+
1
4
≥0"ÄK§ÏLR
v
1(1)-(4)§R
v
2(1)-(2)ÚR
v
3c
0
(f) ≥−3+
4
5
+
4
5
+
7
5
≥0"dé
¡5§d(f
2
)=3†d(f
3
)=3˜"y3=•Äd(f
1
)=3§Xã2(d)¤«§d(f
2
)=d(f
2
)≥6§Ï
LR
v
1(1)§R
v
2(1)-(2)ÚR
v
3(1)c
0
(f) ≥−3+
7
5
+
5
6
+
5
6
≥0"
Figure2.TheconfigurationsusedinTheorem0.2
ã2.½n0.2˜ã
œ/3.n
4
(f) = 1§n
5
+
(f) = 2"Ø”˜„5§-d(v
2
) = 4"bf
3
(f) = 0§ÏLR
v
1(1)-(4)§
R
v
2(1)-(2)§R
v
3(1)ÚR
f
1(2)§kc
0
(f) ≥−3+
7
5
+
4
5
+
7
5
−
3
5
≥0"bf
3
(f) = 1§-d(f
1
) = 3§Xã
2(e) ¤«§d(f
2
) = d(f
3
) ≥6"ÏLR
v
1(1)§R
v
2(1)-(2)ÚR
v
1(1) kc
0
(f) ≥−3+
17
15
+
7
5
+
5
6
≥0"d
é¡5§d(f
2
) =3†d(f
1
) =3˜"bd(f
3
) =3"XJv
1
Úv
3
Ù¥˜‡´5-º:§…†n‡3-¡
ƒ§,˜:´5
+
-º:§Ø”˜„5§-v
3
´Xã2(f)¤«5-¡§ÏLR
v
1(1)§R
v
1(3)-(4)§
R
v
2(1)-(2)§R
v
3(1)ÚR
f
1(1) kc
0
(f) ≥−3+
17
15
+
17
15
+
5
6
−
1
10
≥0"ÄK§ÏLR
v
1(1)§R
v
1(3)-(4)§
DOI:10.12677/pm.2021.1181761597nØêÆ
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R
v
2(1)-(2)§R
v
3(1)ÚR
f
1(2)kc
0
(f) ≥−3+
7
5
+
5
6
+
7
5
−
3
5
≥0"
œ/4.n
5
+
(f) = 3"ÏLR
v
2(1)-(2)ÚR
v
3(1)kc
0
(f) ≥−3+
17
15
+
17
15
+
7
5
≥0"
y3·‚y²Gz‡:vkšKŠ"
-d(v) = 4"ÏLÚn2.2kf
3
(v) ≤2"XJf
3
(v) = 2§ÏLR
v
1(1)-(2)ÚR
v
1(6)-(7)kc
0
(v) ≥
2−max{2×
4
5
+2×
1
5
,2×
5
6
+2×
1
6
,2×
4
5
+
1
6
+
1
5
}≥0"XJf
3
(v)=1§ÏLR
v
1(3)-(7)
kc
0
(v) ≥2−max{
1
2
+
7
6
+2×
1
6
,
7
5
+f
5
(v)×
1
5
+f
6
+
(v)×
1
6
}≥0"XJf
3
(v) = 0§ÏLR
v
1(5)-(7)§
kc
0
(v) ≥2−max{f
4
(v)×
1
2
+f
5
(v)×
1
5
+f
6
+
(v)×
1
6
}≥0"
-d(v)=5"ÏLÚn2.2kf
3
(v)≤3"XJf
3
(v)=3"ÏLR
v
2(1)kc
0
(v)≥4 −
17
15
−
17
15
−
7
5
−2×
1
6
≥0"XJf
3
(v)=2§…ü‡3-¡ƒ§ÏLR
v
2(4)kc
0
(v)≥4−max{2 ×
7
5
+2×
1
6
+
1
2
,2×
7
5
+2×
1
6
+
3
5
,2×
7
5
+3×
1
6
}≥0"XJü‡3-¡Øƒ§ŠâR
v
2(5)-
(8)?رeA«œ/µÏLR
v
2(2)ÚR
v
2(5)1§c
0
(v)=4−
3
5
−
1
5
−
2
5
−2 ×
7
5
=0¶ÏL
R
v
2(2)ÚR
v
2(5)2§c
0
(v)=4−
3
5
−
3
10
−
3
10
−2×
7
5
=0¶ÏLR
v
2(2)ÚR
v
2(6)§kc
0
(v)≥
4−
3
5
−
2
5
−
1
6
−2 ×
7
5
≥0¶ÏLR
v
2(2)ÚR
v
2(7)§kc
0
(v)≥4−
2
5
−
2
5
−
1
6
−2 ×
7
5
≥0¶ÏL
R
v
2(2)ÚR
v
2(8)§kc
0
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3
5
−2×
1
6
−2×
7
5
≥0"XJf
3
(v) = 1§ÏLR
v
2(2)-(3)ÚR
v
2(9)-
(10)§kc
0
(v)≥4−
7
5
−max{f
4
(v) ×
1
2
+ f
5
(v) ×
3
5
+ f
6
+
(v) ×
1
6
}≥0"XJf
3
(v)=0"ÏL
R
v
2(3)ÚR
v
2(9)-(10)§kc
0
(v) ≥4−max{f
4
(v)×
1
2
+f
5
(v)×
3
5
+f
6
+
(v)×
1
6
}≥0"
-d(v)=k(k≥6)"Ï LR
v
3§XJk≥7§Kc
0
(v)>(2k−6) −max{f
3
(v)×
7
5
+ (k−
f
3
(v))×
3
5
}≥0"Ïd=•Äk= 6"XJk= 6§kf
3
(v) ≤4"XJf
3
(v) = 4§ÏLR
v
3(1)ÚR
v
3(4)
c
0
(v) = 6−4×
7
5
−2×
1
6
≥0"XJf
3
(v) ≤3§ÏLR
v
3kc
0
(v) ≥6−max{f
3
(v)×
7
5
+f
4
(v)×
1
2
+
f
5
(v)×
3
5
+f
6
+
(v)×
1
6
}≥0
–d§·‚y²é¤kv∈V(G)Úf∈F(G)Ñ÷v
P
c
0
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c
0
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ë•©z
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