设为首页
加入收藏
期刊导航
网站地图
首页
期刊
数学与物理
地球与环境
信息通讯
经济与管理
生命科学
工程技术
医药卫生
人文社科
化学与材料
会议
合作
新闻
我们
招聘
千人智库
我要投稿
办刊
期刊菜单
●领域
●编委
●投稿须知
●最新文章
●检索
●投稿
文章导航
●Abstract
●Full-Text PDF
●Full-Text HTML
●Full-Text ePUB
●Linked References
●How to Cite this Article
AdvancesinAppliedMathematics
A^
ê
Æ
?
Ð
,2022,11(3),973-979
PublishedOnlineMarch2022inHans.http://www.hanspub.org/journal/aam
https://doi.org/10.12677/aam.2022.113104
´
Ú
Ü
Î
Ò
¦
È
ã
>
/Ú
äää
···
ú
ô
“
‰
Œ
Æ
§
ê
Æ
†
O
Ž
Å
‰
ÆÆ
§
ú
ô
7
u
Â
v
F
Ï
µ
2022
c
2
9
F
¶
¹
^
F
Ï
µ
2022
c
3
4
F
¶
u
Ù
F
Ï
µ
2022
c
3
11
F
Á
‡
2019
c
§
Behr
|
^
Î
Ò
ã
>
/Ú
V
g
y
²
é
u
?
¿
Î
Ò
ã
(
G,σ
)
Ñ
k
∆(
G,σ
)
≤
χ
0
(
G,σ
)
≤
∆(
G,σ
)+1
§
Ù
¥
χ
0
(
G,σ
)
´
(
G,σ
)
>
/Ú
ê
§
∆(
G,σ
)
´
(
G,σ
)
•
Œ
Ý
"
©
·
‚
y
²
3
´
Ú
Ü
Î
Ò
¦
È
ã
(
P
n
2
T
m
,σ
)
¥
§
Ù
¥
P
n
Ú
T
m
©
O
´
k
n
‡
º:
´
Ú
k
m
‡
º:
Ü
§
n>
2
…
∆(
T
m
)
>
1
ž
§
K
χ
0
(
P
n
2
T
m
,σ
) = ∆(
P
n
2
T
m
,σ
)
"
'
…
c
Î
Ò
ã
§
¦
È
ã
§
´
§
Ü
§
>
/Ú
EdgeColoringoftheSignedProduct
GraphsofPathsandForests
YajingWang
CollegeofMathematicsandComputerScienceofZhejiangNormalUniversity,JinhuaZhejiang
Received:Feb.9
th
,2022;accepted:Mar.4
th
,2022;published:Mar.11
th
,2022
Abstract
2019,Behrusedtheconceptofedgecoloringofsignedgraphstoprovethatforany
©
Ù
Ú
^
:
ä
·
.
´
Ú
Ü
Î
Ò
¦
È
ã
>
/Ú
[J].
A^
ê
Æ
?
Ð
,2022,11(3):973-979.
DOI:10.12677/aam.2022.113104
ä
·
signedgraphs
(
G,σ
)
thereis
∆(
G,σ
)
≤
χ
0
(
G,σ
)
≤
∆(
G,σ
)+1
,where
χ
0
(
G,σ
)
isthenumber
ofedgecoloringof
(
G,σ
)
,
∆(
G,σ
)
isthemaximumdegreeof
(
G,σ
)
.Inthispaper,we
provethatinthesignedproductgraphsofpathsandforests
(
P
n
2
T
m
,σ
)
,
P
n
and
T
m
arerespectivelypathswiththenumberof
n
verticesandforestswiththenumberof
m
vertices.When
n>
2
and
∆(
T
m
)
>
1
,then
χ
0
(
P
n
2
T
m
,σ
) = ∆(
P
n
2
T
m
,σ
)
.
Keywords
SignedGraphs,ProductGraphs,Paths,Forests,Edge-Coloring
Copyright
c
2022byauthor(s)andHansPublishersInc.
This work is licensed undertheCreative Commons Attribution InternationalLicense(CCBY4.0).
http://creativecommons.org/licenses/by/4.0/
1.
Ú
ó
Harary
3
þ
-
V
50
c
“
J
Ñ
Î
Ò
ã
V
g
[1]
"
Î
Ò
ã
(
G,σ
)
´
‰
Ä
:
ã
G
=(
V
(
G
)
,E
(
G
))
>
8\
˜
‡
Î
Ò
N
σ
:
E
(
G
)
→{
+1
,
−
1
}
§
¦
G
z
˜
^
>
e
Ñ
k
˜
‡
Î
Ò
σ
(
e
)
"
σ
(
e
)=
+1
ž
§
>
e
•
>
¶
σ
(
e
)=
−
1
ž
§
>
e
•
K
>
"
(
G,σ
)
˜
‡
incidence
´
:
-
>
k
S
é
(
v,e
)
§
v
´
e
˜
‡
à:
"
‰
½
˜
‡
Î
Ò
ã
(
G,σ
)
§
I
(
G
)
L
«
(
G,σ
)
¥
¤
k
incidence
8
Ü
"
V
(
G
)
L
«
(
G,σ
)
º:
8
¶
E
(
G,σ
)
L
«
(
G,σ
)
>
8
"
∆(
G
)
L
«
G
•
Œ
Ý
"
©
¤
?
Ø
Î
Ò
ã
þ
´
Ã
‚
§
Ã
-
>
{
ü
k
•
Î
Ò
ã
"
©™
`
²
P
ÒÚ
â
Š
þ
ë
©
z
[2]
"
½
Â
1
{
ü
Î
Ò
ã
(
G,σ
)
>
/Ú
´
•
é
(
G,σ
)
incidence
X
Ú
§
¦
Ó
˜
‡
º:
'
é
incidence
þ
k
Ø
Ó
ô
Ú
"
Î
Ò
ã
(
G,σ
)
>
Ú
ê
χ
0
(
G,σ
)
´
•
¦
Î
Ò
ã
(
G,σ
)
•
~
>
/Ú
•
ô
Ú
ê
"
Vizing
3
1964
c
§
‰
Ñ
{
ü
ã
>
/Ú
ê
þ
e
.
"
½
n
1(Vizing,1964,[3])
˜
‡
ã
G
>
/Ú
ê
χ
0
(
G
)
÷
v
∆(
G
)
≤
χ
0
(
G
)
≤
∆(
G
)+1
"
2019
c
§
Behr
‰
Ñ
{
ü
Î
Ò
ã
>
/Ú
V
g
[4]
¿
y
²
þ
ã
(
Ø
3
{
ü
Î
Ò
ã
(
G,σ
)
¥
Ó
÷
v
∆(
G,σ
)
≤
χ
0
(
G,σ
)
≤
∆(
G,σ
)+1
"
½
Â
2
ã
G
Ú
H
¦
È
ã
G
2
H
´
•
ä
k
º:
8
V
(
G
2
H
)
Ú
>
8
E
(
G
2
H
)
ã
§
Ù
¥
V
(
G
2
H
) =
V
(
G
)
×
V
(
H
)=
{
(
u,v
)
|
u
∈
V
(
G
)
,v
∈
V
(
H
)
}
,
E
(
G
2
H
)=
{
(
u,v
)(
u
0
,v
0
)
|
u
=
u
0
,vv
0
∈
E
(
H
)
½
v
=
v
0
,uu
0
∈
E
(
G
)
}
"
DOI:10.12677/aam.2022.113104974
A^
ê
Æ
?
Ð
ä
·
½
Â
3
Î
Ò
¦
È
ã
(
G
2
H,σ
)
´
‰
§
>
8\
˜
‡
Î
Ò
N
σ
:
E
(
G
2
H
)
→{
+1
,
−
1
}
§
¦
G
2
H
z
˜
^
>
e
Ñ
k
˜
‡
Î
Ò
σ
(
e
)
"
½
Â
4
é
u
Î
Ò
ã
(
G,σ
)
§
X
J
χ
0
(
G,σ
)=∆(
G,σ
)
§
K
¡
Î
Ò
ã
(
G,σ
)
•
1
˜
a
Î
Ò
ã
"
P
(
G,σ
)
∈C
1
"
½
Â
5
é
u
Î
Ò
ã
(
G,σ
)
§
X
J
χ
0
(
G,σ
)=∆(
G,σ
) +1
§
K
¡
Î
Ò
ã
(
G,σ
)
•
1
a
Î
Ò
ã
"
P
(
G,σ
)
∈C
2
"
3
©
z
[5]
§
[6]
§
[7]
¥
§
þ
ï
Ä
Î
Ò
ã
/Ú
"
3
©
¥
§
·
‚
ò
Î
Ò
ã
˜
‡
n
-
>
/Ú
½
Â
•
:
½
Â
6
(
G,σ
)
´
˜
‡
Î
Ò
ã
§
n
´
˜
‡
ê
§
e
n
=2
k
§
K
(
G,σ
)
˜
‡
n
-
>
/
Ú
Ò
´
˜
‡
N
γ
:
E
(
G
)
→{±
k,
±
(
k
−
1)
,
···
,
±
1
}
¶
e
n
=2
k
+ 1
§
K
γ
:
E
(
G
)
→{±
k,
±
(
k
−
1)
,
···
,
±
1
,
0
}
§
¦
é
?
¿
>
e
=
vw
∈
E
(
G
)
Ñ
k
γ
(
v,e
)=
σ
(
e
)
γ
(
w,e
)
"
?
¿
ƒ
ü
^
>
e
1
=
vw
Ú
e
2
=
vu
§
¦
γ
(
v,e
1
)
6
=
γ
(
v,e
2
)
§
K
¡
γ
´
(
G,σ
)
þ
~
n-
>
/Ú
"
Behr
Ï
L
é
½
n
1
?
1
í
2
§
?
¿
{
ü
Î
Ò
ã
(
G,σ
)
>
/Ú
ê
χ
0
(
G,σ
)
÷
v
∆(
G,σ
)
≤
χ
0
(
G,σ
)
≤
∆(
G,σ
)+ 1
"
3
©
¥
§
·
‚
y
²
´
Ú
Ü
Î
Ò
¦
È
ã
(
P
n
2
T
m
,σ
)
>
/Ú
ê
χ
0
(
P
n
2
T
m
,σ
)
÷
v
n>
2
…
∆(
T
m
)
>
1
ž
§
K
χ
0
(
P
n
2
T
m
,σ
)=∆(
P
n
2
T
m
,σ
)
"
·
‚
©
¤
2
‡
½
n5
y
²
ù
‡
(
Ø
"
·
‚
Ä
k
y
²
3
´
Ú
Ü
Î
Ò
¦
È
ã
(
P
n
2
T
m
,σ
)
¥
§
1
≤
∆(
T
m
)
≤
2
ž
§
½
n
2
¤
á
¶
2
y
²
n>
2
…
∆(
T
m
)
>
2
ž
§
½
n
3
¤
á
"
½
n
2
3
Î
Ò
¦
È
ã
(
P
n
2
T
m
,σ
)
¥
§
Ù
¥
n
§
m
©
O
´
P
n
§
T
m
þ
:
ê
"
(
i
)
n
= 2
§
∆(
T
m
) =1
…
Î
Ò
ã
(
P
n
2
T
m
,σ
)
´
˜
‡
š
²
ï
ž
§
K
Î
Ò
ã
(
P
n
2
T
m
,σ
)
•
1
a
Î
Ò
ã
"
(
ii
)
n
= 2
§
∆(
T
m
) =1
…
Î
Ò
ã
(
P
n
2
T
m
,σ
)
´
˜
‡
²
ï
ž
§
K
Î
Ò
ã
(
P
n
2
T
m
,σ
)
•
1
˜
a
Î
Ò
ã
"
(
iii
)
n>
2
…
∆(
T
m
) = 2
ž
§
K
Î
Ò
ã
(
P
n
2
T
m
,σ
)
•
1
˜
a
Î
Ò
ã
"
½
n
3
3
Î
Ò
¦
È
ã
(
P
n
2
T
m
,σ
)
¥
§
Ù
¥
n
§
m
©
O
´
P
n
§
T
m
þ
:
ê
"
n>
2
…
∆(
T
m
)
>
2
ž
§
K
Î
Ò
ã
(
P
n
2
T
m
,σ
)
•
1
˜
a
Î
Ò
ã
"
2.
½
n
2
y
²
y
²
3
Î
Ò
¦
È
ã
(
P
n
2
T
m
,σ
)
¥
§
1
≤
∆(
T
m
)
≤
2
ž
§
Î
Ò
ã
(
P
n
2
T
m
,σ
)=(
P
n
2
P
m
,σ
)
"
d
P
n
2
P
m
(
•
§
(
P
n
2
P
m
,σ
)
´
‘
k
Î
Ò
•
‚
ã
"
Ä
k
y
²
(
i
)
"
n
=2
§
∆(
T
m
)=1
…
Î
Ò
ã
(
P
n
2
T
m
,σ
)
•
k
˜
‡
š
²
ï
ž
·
‚
ù
‡
š
²
ï
•
C
=
v
0
v
1
v
2
v
3
v
0
"
v
3
v
0
´
˜
^
>
§
Ï
•
C
´
š
²
ï
§
¤
±
P
=
v
0
v
1
v
2
v
3
þ
˜
½
k
Û
ê
^
K
>
"
P
–
Œ
±
^
2
«
ô
Ú/
Ð
§
^
±
a
/
Ð
"
b
l
(
v
0
,v
0
v
1
)
m
©
^
a
Ú
−
a
?
1
/Ú
§
K
(
v
3
,v
2
v
3
)
˜
½
/
−
a
"
¤
±
C
þ
•
˜
^
>
v
3
v
0
˜
½
‡
^
1
n
«
ô
Ú
â
U
/
Ð
§
•
b
"
¤
±
χ
0
(
P
2
2
P
2
,σ
) = 3
"
d
ž
Î
Ò
ã
∆(
P
2
2
P
2
,σ
) = 2
§
¤
±
d
ž
Î
Ò
ã
(
P
n
2
T
m
,σ
)
´
1
a
Î
Ò
ã
"
DOI:10.12677/aam.2022.113104975
A^
ê
Æ
?
Ð
ä
·
Ù
g
y
²
(
ii
)
"
n
=2
§
∆(
T
m
)=1
…
Î
Ò
ã
(
P
n
2
T
m
,σ
)
•
k
˜
‡
²
ï
ž
·
‚
ù
‡
²
ï
•
C
0
=
v
0
0
v
0
1
v
0
2
v
0
3
v
0
0
"
v
0
3
v
0
0
´
˜
^
>
§
Ï
•
C
0
´
²
ï
§
¤
±
P
0
=
v
0
0
v
0
1
v
0
2
v
0
3
þ
˜
½
k
ó
ê
^
K
>
"
P
0
–
Œ
±
^
2
«
ô
Ú/
Ð
§
^
±
a
/
Ð
"
b
l
(
v
0
0
,v
0
0
v
0
1
)
m
©
^
a
Ú
−
a
?
1
/Ú
§
K
(
v
0
3
,v
0
2
v
0
3
)
˜
½
/
a
"
¤
±
C
0
þ
•
˜
^
>
v
0
3
v
0
0
Œ
±
^
−
a
/
Ð
"
¤
±
χ
0
(
P
2
2
P
2
,σ
) = 2
"
d
ž
Î
Ò
ã
∆(
P
2
2
P
2
,σ
) = 2
§
¤
±
Î
Ò
ã
(
P
n
2
T
m
,σ
)
´
1
˜
a
Î
Ò
ã
"
•
·
‚
y
²
(
iii
)
"
n>
2
…
∆(
T
m
) = 2
ž
§
∆(
P
n
2
P
m
) = 4
"
·
‚
‰
Ñ
ù
‡
Î
Ò
ã
(
P
n
2
P
m
,σ
)
˜
‡
~
4-
>
/Ú
γ
¿
…
y
²
ù
‡
.
´
;
"
Ä
k
·
‚
‰
Ñ
ù
‡
Î
Ò
ã
(
P
n
2
P
m
,σ
)
˜
‡
~
4-
>
/Ú
γ
"
é
u
?
¿
i
§
j
§
i
∈
[
m
]
§
j
∈
[
n
]
§
i
L
«
(
P
n
2
P
m
,σ
)
1
i
1
§
j
L
«
(
P
n
2
P
m
,σ
)
1
j
§
u
i,j
L
«
(
P
n
2
P
m
,σ
)
þ
1
i
1
§
1
j
:
"
-
γ
(
u
i,
1
,e
(
u
i,
1
u
i,
2
)) = 1
,
γ
(
u
i,j
,e
(
u
i,j
u
i,j
−
1
)) =
γ
(
u
i,j
−
1
,e
(
u
i,j
−
1
u
i,j
))
σ
(
e
(
u
i,j
−
1
u
i,j
))
,
γ
(
u
i,j
,e
(
u
i,j
u
i,j
+1
)) =
−
γ
(
u
i,j
,e
(
u
i,j
u
i,j
−
1
))
,
(1)
-
E
þ
ã
/Ú
L
§
,
†
3
1
m
1
"
é
u
?
¿
1
i
1
´
P
i
5
`
§
§
•
˜
^
>
e
(
u
i,n
−
1
u
i,n
)
/
Ú
•
:
γ
(
u
i,n
−
1
,e
(
u
i,n
−
1
u
i,n
−
2
)) =
γ
(
u
i,n
−
2
,e
(
u
i,n
−
1
u
i,n
−
2
))
σ
(
e
(
u
i,n
−
2
u
i,n
−
1
))
,
γ
(
u
i,n
−
1
,e
(
u
i,n
−
1
u
i,n
)) =
−
γ
(
u
i,n
−
1
,e
(
u
i,n
−
2
u
i,n
−
1
))
,
γ
(
u
i,n
,e
(
u
i,n
u
i,n
−
1
)) =
γ
(
u
i,n
−
1
,e
(
u
i,n
−
1
u
i,n
))
σ
(
e
(
u
i,n
−
1
u
i,n
))
,
(2)
d
i
?
¿
5
•
(
P
n
2
P
m
,σ
)
z
˜
1
incidence
Ñ
Œ
±
^
?
¿
2
‡
ƒ
‡
ô
Ú
±
a
/
Ð
"
þ
ã
/Ú
·
‚
´
a
= 1
"
;
X
-
γ
(
u
1
,j
,e
(
u
1
,j
u
2
,j
)) = 2
,
γ
(
u
i,j
,e
(
u
i,j
u
i
−
1
,j
)) =
γ
(
u
i
−
1
,j
,e
(
u
i
−
1
,j
u
i,j
))
σ
(
e
(
u
i
−
1
,j
u
i,j
))
,
γ
(
u
i,j
,e
(
u
i,j
u
i
+1
,j
)) =
−
γ
(
u
i,j
,e
(
u
i,j
u
i
−
1
,j
))
,
(3)
-
E
þ
ã
/Ú
L
§
,
†
3
1
n
"
é
u
?
¿
1
j
´
P
j
5
`
§
§
•
˜
^
>
e
(
u
m
−
1
,j
u
m,j
)
/Ú
•
:
γ
(
u
m
−
1
,j
,e
(
u
m
−
1
,j
u
m
−
2
,j
)) =
γ
(
u
m
−
2
,j
,e
(
u
m
−
1
,j
u
m
−
2
,j
))
σ
(
e
(
u
m
−
1
,j
u
m
−
2
,j
))
,
γ
(
u
m
−
1
,j
,e
(
u
m
−
1
,j
u
m,j
)) =
−
γ
(
u
m
−
1
,j
,e
(
u
m
−
2
,j
u
m
−
1
,j
))
,
γ
(
u
m,j
,e
(
u
m,j
u
m
−
1
,j
)) =
γ
(
u
m
−
1
,j
,e
(
u
m
−
1
,j
u
m,j
))
σ
(
e
(
u
m
−
1
,j
u
m,j
))
,
(4)
d
j
?
¿
5
•
(
P
n
2
P
m
,σ
)
z
˜
incidence
Ñ
Œ
±
^
Ø
Ó
u
±
a
?
¿
2
‡
ƒ
‡
ô
Ú
±
b
/
Ð
"
þ
ã
/Ú
·
‚
´
b
= 2
"
DOI:10.12677/aam.2022.113104976
A^
ê
Æ
?
Ð
ä
·
q
Ï
•
P
n
2
P
m
?
¿
2
1
Ú
?
¿
2
Ñ
Ø
¬
u
˜
:
"
¤
±
(
P
n
2
P
m
,σ
)
Œ
±
^
4
‡
ô
Ú
±
a
Ú
±
b
/
Ð
"
=
γ
´
Î
Ò
ã
(
P
n
2
P
m
,σ
)
˜
‡
~
4-
>
/Ú
§
χ
0
(
P
n
2
P
m
,σ
) = 4
"
d
Behr
{
ü
Î
Ò
ã
>
/Ú
(
Ø
§
·
‚
•
é
u
?
¿
{
ü
Î
Ò
ã
(
G,σ
)
§
χ
0
(
G,σ
)
e
.
´
∆(
G,σ
)
"
·
‚
3
þ
¡
y
²
¥
®
²
‰
Ñ
1
«
/Ú
•{
χ
0
(
P
n
2
P
m
,σ
)=4
§
¤
±
χ
0
(
P
n
2
P
m
,σ
)=∆(
P
n
2
P
m
,σ
)
§
=
(
P
n
2
P
m
,σ
)
´
1
˜
a
"
n>
2
…
∆(
T
m
)=2
ž
§
T
m
=
P
m
§
¤
±
Î
Ò
ã
(
P
n
2
T
m
,σ
)
•
1
˜
a
Î
Ò
ã
"
3.
½
n
3
y
²
y
²
é
u
Î
Ò
¦
È
ã
(
P
n
2
T
m
,σ
)
5
`
§
n>
2
…
∆(
T
m
)
>
2
ž
§
·
‚
‰
Ñ
ù
‡
Î
Ò
ã
˜
‡
~
∆(
P
n
2
T
m
,σ
)-
>
/Ú
γ
¿
…
y
²
ù
‡
.
´
;
"
d
P
n
2
T
m
(
•
§
n>
2
…
∆(
T
m
)
>
2
ž
§
∆(
P
n
2
T
m
)=∆(
T
m
)+2
"
ë
•
½
n
2
/Ú
•{
§
a
q
/
"
·
‚
k
‰
(
P
n
2
T
m
,σ
)
z
˜
1
?
1
/Ú
"
(
P
n
2
T
m
,σ
)
z
˜
1
þ
´
:
Ø
´
P
i
(
i
∈
[
m
])
§
Ï
d
z
‡
P
i
þ
Œ
±
^
2
«
ô
Ú
±
a
/
Ð
"
q
Ï
•
(
P
n
2
T
m
,σ
)
z
˜
þ
´
:
Ø
ä
T
j
(
j
∈
[
n
])
"
Ï
•
Š
â
©
z
[2]
¥
·
K
2
.
2:
X
J
Î
Ò
ã
´
˜
‡
•
Œ
Ý
•
∆
ä
T
§
K
χ
0
(
T,σ
) = ∆
"
Œ
•
é
u
?
¿
˜
†
ä
T
m
þ
÷
v
χ
0
(
T
m
,σ
)=∆(
T
m
)
"
q
Ï
•
T
j
´
üü
:
Ø
ƒ
§
¤
±
z
‡
T
j
þ
Œ
±
^
Ø
Ó
u
±
a
∆(
T
m
)
«
ô
Ú/
Ð
"
=
χ
0
(
P
n
2
T
m
,σ
) =∆(
T
m
)+2 =∆(
P
n
2
T
m
)
"
¤
±
§
Î
Ò
ã
(
P
n
2
T
m
,σ
)
•
1
˜
a
Î
Ò
ã
"
Ï
L
½
n
2
§
½
n
3
y
²
§
·
‚
Œ
±
µ
´
Ú
Ü
Î
Ò
¦
È
ã
(
P
n
2
T
m
,σ
)
>
/Ú
ê
χ
0
(
P
n
2
T
m
,σ
)
÷
v
n>
2
…
∆(
T
m
)
>
1
ž
§
K
χ
0
(
P
n
2
T
m
,σ
) = ∆(
P
n
2
T
m
,σ
)
"
3
Î
Ò
ã
(
P
n
2
P
m
,σ
)
¥
§
n>
2
§
m
=2
½
n
=2
§
m>
2
ž
§
K
Î
Ò
ã
(
P
n
2
P
m
,σ
)=
(
P
n
2
P
2
,σ
)
½
(
P
n
2
P
m
,σ
)=(
P
2
2
P
m
,σ
)
"
3
(
P
n
2
T
m
,σ
)
¥
§
n>
2
§
∆(
T
m
)=1
ž
§
K
Î
Ò
ã
(
P
n
2
T
m
,σ
) = (
P
n
2
P
2
,σ
)
"
¯¢
þ
n
=
m
ž
§
(
P
n
2
P
2
,σ
)
Ú
(
P
2
2
P
m
,σ
)
´
Ó
˜
‡
{
ü
Î
Ò
ã
"
é
u
(
P
n
2
P
2
,σ
)
5
`
§
n>
2
ž
§
K
χ
0
(
P
n
2
P
2
,σ
)
Œ
U
´
∆(
P
n
2
P
2
,σ
)+1
§
„
ã
1
§
•
Œ
U
´
∆(
P
n
2
P
2
,σ
)
§
„
ã
2
"
ã
1
§
ã
2
´
þ
´
(
P
n
2
P
2
,σ
)
f
ã
"
Figure1.
χ
0
(
P
3
2
P
2
,σ
) = ∆(
P
n
2
P
2
,σ
)+1
ã
1.
χ
0
(
P
3
2
P
2
,σ
) = ∆(
P
n
2
P
2
,σ
)+1
DOI:10.12677/aam.2022.113104977
A^
ê
Æ
?
Ð
ä
·
Figure2.
χ
0
(
P
3
2
P
2
,σ
) = ∆(
P
n
2
P
2
,σ
)
ã
2.
χ
0
(
P
3
2
P
2
,σ
) = ∆(
P
n
2
P
2
,σ
)
3
ã
1
¥
§
u
2
,
2
'
é
2
^
K
>
e
(
u
2
,
1
u
2
,
2
)
§
e
(
u
2
,
3
u
2
,
2
)
§
1
^
>
e
(
u
1
,
2
u
2
,
2
)
§
¿
…
2
‡
ú
>
´
>
e
(
u
1
,
2
u
2
,
2
)
§
d
ž
u
2
,
2
'
é
ù
3
^
>
–
‡
^
3
«
ô
Ú/
Ð
"
3
ã
1
¥
§
-
γ
(
u
1
,
2
,e
(
u
1
,
2
u
2
,
2
)) = 2
,
γ
(
u
2
,
2
,e
(
u
1
,
2
u
2
,
2
)) = 2
,
γ
(
u
2
,
2
,e
(
u
2
,
2
u
2
,
1
)) =
−
1
,
γ
(
u
2
,
1
,e
(
u
2
,
2
u
2
,
1
)) = 1
,
γ
(
u
2
,
2
,e
(
u
2
,
2
u
2
,
3
)) = 1
,
γ
(
u
2
,
3
,e
(
u
2
,
2
u
2
,
3
)) =
−
1
.
(5)
‡
Ž
ã
1
¤
^
ô
Ú
•
§
•
U
-
γ
(
u
2
,
1
,e
(
u
2
,
1
u
1
,
1
)) =
−
1
,
γ
(
u
1
,
1
,e
(
u
2
,
1
u
1
,
1
)) = 1
,
γ
(
u
2
,
3
,e
(
u
2
,
3
u
1
,
3
)) = 1
,
γ
(
u
1
,
3
,e
(
u
2
,
3
u
1
,
3
)) =
−
1
,
γ
(
u
1
,
3
,e
(
u
1
,
3
u
1
,
2
)) = 1
,
γ
(
u
1
,
2
,e
(
u
1
,
3
u
1
,
2
)) =
−
1
.
(6)
d
ž
§
γ
(
u
1
,
1
,e
(
u
1
,
1
u
1
,
2
))
§
γ
(
u
1
,
2
,e
(
u
1
,
1
u
1
,
2
))
•
U
^
1
o
«
ô
Ú
â
U
/
Ð
"
¤
±
§
ã
1
¥
(
P
3
2
P
2
,σ
)
>
Ú
ê
χ
0
(
P
3
2
P
2
,σ
)
´
∆(
P
n
2
P
2
,σ
)+1
"
Ï
d
§
é
u
(
P
n
2
P
2
,σ
)
§
k
˜
Ü
©Î
Ò
¦
È
ã
÷
v
χ
0
(
P
n
2
P
2
,σ
)=∆(
P
n
2
P
2
,σ
)+ 1
´
1
˜
a
§
•
k
˜
Ü
©Î
Ò
¦
È
ã
÷
v
χ
0
(
P
n
2
P
2
,σ
)=∆(
P
n
2
P
2
,σ
)
´
1
a
§
ù
„
I
‡
·
‚
?
˜
Ú
ï
Ä
"
DOI:10.12677/aam.2022.113104978
A^
ê
Æ
?
Ð
ä
·
ë
•
©
z
[1]Harary,F.(1953)OntheNotionofBalanceofaSignedGraph.
MichiganMathematicalJournal
,
2
,143-146.https://doi.org/10.1307/mmj/1028989917
[2]Zhang, L., Lu, Y., Lou, R., Ye, D. and Zhang, S.(2020) EdgeColoring SignedGraphs.
Discrete
AppliedMathematics
,
282
,234-242.https://doi.org/10.1016/j.dam.2019.12.004
[3]Vizing,V.(1964)OnanEstimateoftheChromaticClassofap-Graph.
DiscretAnaliz
,
3
,
23-30.
[4]Behr,R.(2020) EdgeColoringSigned Graphs.
DiscreteMathematics
,
343
,ArticleID:111654.
https://doi.org/10.1016/j.disc.2019.111654
[5]Cartwright,D.andHarary,F.(1968)OntheColoringofSignedGraphs.
ElementederMath-
ematik
,
23
,85-89.
[6]Zaslavsky,T.(1991)OrientationofSignedGraphs.
EuropeanJournalofCombinatorics
,
12
,
361-375.https://doi.org/10.1016/S0195-6698(13)80118-7
[7]Zaslavsky,T.(1982)SignedGraphs.
DiscreteAppliedMathematics
,
4
,47-74.
https://doi.org/10.1016/0166-218X(82)90033-6
DOI:10.12677/aam.2022.113104979
A^
ê
Æ
?
Ð