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AdvancesinAppliedMathematicsA^êÆ?Ð,2022,11(1),318-325
PublishedOnlineJanuary2022inHans.http://www.hanspub.org/journal/aam
https://doi.org/10.12677/aam.2022.111039
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TheGeneralizedColoringNumber
andGameColoringNumberof
ProductGraphofTree
andPath
JialiLiu
CollegeofMathematicsandComputerScience,ZhejiangNormalUniversity,JinhuaZhejiang
Received:Dec.24
th
,2021;accepted:Jan.19
th
,2022;published:Jan.26
th
,2022
©ÙÚ^:4Zw.äÚ´¦Èã2Â/Úê9Ɖ/Úê[J].A^êÆ?Ð,2022,11(1):318-325.
DOI:10.12677/aam.2022.111039
4Zw
Abstract
Thispaperconsiderstheproductgraphofsimplegraphtreeandpath,givesalinear
orderoftheproductgraphoftreeandpath,andintroducesthegeneralizedcoloring
numberoftheproductgraphoftreeandpath.Meanwhile,wegiveanorientationthat
themaximumout-degreeoftheproductgraphoftreeandpathisatmostaconstant
andintroducethegamecoloringnumberoftheproductgraphoftreeandpath.
Keywords
ProductGraph,GameColoringNumber,GeneralizedColoringNumber
Copyright
c
2022byauthor(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/aam.2022.111039320A^êÆ?Ð
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DOI:10.12677/aam.2022.111039321A^êÆ?Ð
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DOI:10.12677/aam.2022.111039322A^êÆ?Ð
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0
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p(l−k≤p≤l,l≥k),…y
0
≺
L
v,Ïd,
lvfk-Œˆ:31l–õ•k,
lvfk-Œˆ:31l−1 –õ•1+2k,
lvfk-Œˆ:31l−2 –õ•1+2k,(k≥2) ,
,...,
lvfk-Œˆ:31l−i–õ•1+2k,(k≥i) ,
Ïd,·‚kwcol
k
(G) ≤k+(1+2k)k+1 .n:wcol
k
(G) ≤2k
2
+k+1.
dãG©Œ•µ|i−j|≥2ž,1i:†1j:vk>ƒë,Ïd,éu?¿˜
‡:v∈G5`,k≥2ž,lvk-Œˆ:y
0
•Uá3:v¤3þ(Ø”•1l)9:
vþ˜(Ø”•1l−1),…y
0
≺
L
v,d½ÂŒ•G¥:3Ó˜k>,…éu?¿˜:
(x
i,j
,y
m
) –õ•kü‡Ø•(x
i,j
,y
m−1
) ,(x
i,j
,y
m+1
) ,Ïdk>2 ž,lv k-Œˆ:31l
–õ•k,lvk-Œˆ:31l−1 –õ•k+2 ,Ïd,·‚kcol
k
(G) ≤2k+3 .
3.äÚ´¦ÈãƉ/Úê
-T´˜†ä,P´˜^´.éuäTÚ´P¦ÈãGƉ/Úê,·‚ÏL‰ãG˜
‡Ü·½•,¦ãG•ŒÑÝ•˜‡~ê,2(ÜÚn1 ÑãG(a,1) -Ɖ/Úê.
½n4éu?¿êa≥2XJG=T×P,Ù¥T´˜†ä,P´˜^´,@o
(a,1)-col
g
(G)≤6.
y²éuãG:,·‚†æ^½n1 üSÚ©.
e5·‚éãG>?1½•,dT×P½Â±9·‚éT×P¥:©´•,ãT×P
vkÓ˜>.éuƒ>©•ü«œ¹µ
1.e= ((x
i−1,a
,y
m−1
),(x
i,j
,y
m
))••l(x
i,j
,y
m
)(x
i−1,a
,y
m−1
).
2.e= ((x
i−1,a
,y
m+1
),(x
i,j
,y
m
))••l(x
i,j
,y
m
)(x
i−1,a
,y
m+1
).
Ïd,é?¿˜‡:(x
i,j
,y
m
)Ñkd
+
G
((x
i,j
,y
m
)) ≤2ŠâÚn1 Œ(a,1)-col
g
(G)≤6.
½n5éu?¿êa≥2XJG=T2P,Ù¥T´˜†ä,P´˜^´,@o
(a,1)-col
g
(G)≤6.
y²éuãG:,·‚†æ^½n1 üSÚ©.
e5·‚éãG>?1½•,éuÓ˜>e= ((x
i,j
,y
m−1
),(x
i,j
,y
m
))••l(x
i,j
,y
m
)
DOI:10.12677/aam.2022.111039323A^êÆ?Ð
4Zw
(x
i,j
,y
m−1
) .éuƒ>e= ((x
i−1,a
,y
m
) ,(x
i,j
,y
m
)) ••l(x
i,j
,y
m
) (x
i−1,a
,y
m
) .Ï
d,é?¿˜‡:(x
i,j
,y
m
)Ñkd
+
G
((x
i,j
,y
m
)) ≤2ŠâÚn1 Œ(a,1)-col
g
(G)≤6.
½n6éu?¿êa≥4,XJG=TP,Ù¥T´˜†ä,P´˜^´,@o
(a,1)-col
g
(G)≤10.
y²éuãG:,·‚†æ^½n1 üSÚ©.
e5·‚éãG>?1½•,éuÓ˜>e= ((x
i,j
,y
m−1
),(x
i,j
,y
m
))••l(x
i,j
,y
m
)
(x
i,j
,y
m−1
).éuƒ>©•n«œ¹µ
1.e= ((x
i−1,a
,y
m
),(x
i,j
,y
m
))••l(x
i,j
,y
m
)(x
i−1,a
,y
m
).
2.e= ((x
i−1,a
,y
m−1
),(x
i,j
,y
m
))••l(x
i,j
,y
m
)(x
i−1,a
,y
m−1
).
3.e= ((x
i−1,a
,y
m+1
),(x
i,j
,y
m
))••l(x
i,j
,y
m
)(x
i−1,a
,y
m+1
).
Ïd,é?¿˜‡:(x
i,j
,y
m
)Ñkd
+
G
((x
i,j
,y
m
)) ≤4ŠâÚn1 Œ(a,1)-col
g
(G)≤10.
éua<4 ž,(ܽn1½•,Œ±±eíص
íØ1éu?¿êa<4XJG= TP,Ù¥T´˜†ä,P´˜^´,-r
~
G
(
P
) = r,
@o(a,1)−col
g
(G)≤4+b(1+
1
a
)rc+2 .
y²d½n3Œ•é?¿˜‡:(x
i,j
,y
m
),d
+
G
((x
i,j
,y
m
))≤4,ŠâÚn2Œ(a,1)−
col
g
(G)≤4+b(1+
1
a
)rc+2 .
4.(Š
©‰ÑäÚ´¦Èã2Â/Úêþ.±9(a,b) -Ɖ/Úêþ..¿3y²L§
¥‰ÑäÚ´¦Èã˜‡‚5S±9•ŒÑÝ• ›•˜‡~ê½•,3ïÄ,ãëê
ž,I‡^ã•ŒÑÝ.T½•JøïÄg´.3ãØ¥,¦Èã2Â/ÚêÚƉ/Úê
ïÄ,ØÓã¦Èã2Â/Úê9Ɖ/ÚêÑ´Š?˜Ú&?¯K.©‰
ÑäÚ´¦Èã±9¦‚2Â/Úê,?Ø¦ÈãƉ/Úê,@oéu?¿ü‡ã
¦Èã2Â/Úê´Äk˜‡‚5þ.Q,ù•´Š?˜Ú&?¯K.
ë•©z
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20,255-264.https://doi.org/10.1023/B:ORDE.0000026489.93166.cb
[2]Bodlaender,H.L.(1991)OntheComplexityofSomeColoringGames.In:M¨ohring,R.H.,
Eds.,Graph-TheoreticConceptsinComputerScience.WG1990.LectureNotesinComputer
Science,Springer,Berlin,30-40.https://doi.org/10.1007/3-540-53832-129
[3]Kierstead,H.A.(2005)AsymmetricGraphColoringGames.JournalofGraphTheory,48,
169-185.https://doi.org/10.1002/jgt.20049
DOI:10.12677/aam.2022.111039324A^êÆ?Ð
4Zw
[4]Kierstead,H.A.andYang,D.(2005)VeryAsymmetricMarkingGames.Order,22,93-107.
https://doi.org/10.1007/s11083-005-9012-y
[5]Yang,D.andZhu,X.(2008)ActivationStrategyforAsymmetricMarkingGames.European
JournalofCombinatorics,29,1123-1132.https://doi.org/10.1016/j.ejc.2007.07.004
DOI:10.12677/aam.2022.111039325A^êÆ?Ð

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