NAOSITE: Nagasaki University's Academic Output SITE
Title
PERFECT ONE-FACTORIZATIONS OF THE COMPLETE GRAPH
Author(s)
Kobayashi, Midori
Citation
経済学部研究年報, 4, pp.85-90; 1988
Issue Date
1988-02
URL
http://hdl.handle.net/10069/26123
Right
This document is downloaded at: 2017-07-28T16:41:31Z
http://naosite.lb.nagasaki-u.ac.jp
85
PERFECT
ONE-FACTORIZATIONS
THE COMPLETE
MIDORI
OF
GRAPH
KOBAYASHI
1. Introduction
We denote
set of In
pairwise
by K2n=(V,
vertices
E) the complete
and E is the set of n(2n-l)
disjoint
edges
that
partition
is a set of 1-factors
that
perfect
of every
if the
union
1-factorizations
sends
graph
partition
edges.
2n vertices,
A 1-factor
the set of vertices
the
pair
with
set
of edges
of distinct
if there
V is the
of K2n is a set of
V. A 1-factorization
of K2n
E. A 1-factorization
1-factors
F and F" are isomorphic
where
is a Hamiltonian
exists
is
called
circuit.
a permutation
Two
of
V which
each member of F into a member of F.
The existence
is conjectured,
of a perfect
1-factorization
and the problem
16, 28, 244, 344.
Perfect
In this
paper,
1-factorizations
The papers
is settled
these
of the complete
graph
only for 2n=p+l,2p
perfect
1-factorizations
K2n for all n^2
(p is prime),
are explicitly
of K3e, Kl332 and K6860 have recently
and 2n=
shown.
been found
([4,
5]).
are in submission.
2. Perfect
GF{p)
1-factorization
denotes
the Galois
V=GF(p)
U {00}
of KP+\ (p is an odd prime)
field
with
p elements.
We put
and
F
Fo is called
0={{iJ]
a starter
t
+J=O,
1-factor
iJ
e
GF(p)-{0}}
U {(0,oo)J
and
Fs=F0+g
=\{i+gj+g)
is an induced
zation
GKp+i :
1-factor,
i+J=O,
hi*
where g is an element
GF(p)-{0}}
of GF(p).
U {(*,oo)|.
We obtain
a perfect
1-factori-
8
6
ι
{I
gε GF(扮
p+1=
GK
Forexample，as
t
a
r
t
e
ro
fGK
sshowni
nFIGURE1
.
12 i
。
1
0
9
00
as
t
a
r
t
e
ro
fGK
12
FIGURE1
3
.P
e
r
f
e
c
tl
f
a
c
t
o
r
i
z
a
t
i
o
no
fK2p (pi
sanoddprime)
L
e
t
V
=
{
w
げ
w
r，w
t
，
IFJP+I
d
1
)
，
Formathematicals
i
m
p
l
i
c
i
t
y，weu
s
e W川 ρand W
f
+kP i
n
s
t
e
a
do
f Wiand w~
r
e
s
p
e
c
t
i
v
e
l
y，where k i
sani
n
t
e
g
e
r
.
i
t
h0豆 S豆 ρ-1，wep
u
t
Forani
n
t
e
g
e
rsw
列
)
OG
{Wi，W
j
}
1i
十 j三 s
，i
辛 j
(mod
s= {
u{{w~ wj
}I
i+j ρ-2-s，件
三
列
)
j(mod
u{{
ω8/2，Wip-2-S)/2}}，
where1
/
2means2
-1 (modρ
)
.Forani
n
t
e
g
e
rsw
i
t
h0豆 S壬ρ-2，wep
u
t
ι={
{
W
i，wj}I
i+j
三
が
)
s(mod
8
7
Then
{
O
Gs=川 ， ρ-1}U
{
ι
¥s=い ， ト 2
}
ω2ρ=
s¥
i
sap
e
r
f
e
c
t1
f
a
c
t
o
r
i
z
a
t
i
o
no
fK2P((2
)
)
.Forexample，GA
sshowni
nFIGURE2
.
lO i
2
GAlO
FIGURE2
L
e
t
v={vo，
Vl，'
"
，ω
小
E'={
{Vi，
Vj}¥ 0豆 己 2ρ-1
，0豆 信 2ρ-1，日
j
}
Form
a
t
h
e
m
a
t
i
c
a
ls
i
m
p
l
i
c
i
t
y，weu
s
e Vi+2ρk i
n
s
t
e
a
do
f Vi，whereki
sani
n
t
e
g
e
r
.
e
f
i
n
e Gs(cE)a
sf
o
l
l
o
w
s
:
Foranyi
n
t
e
g
e
rsw
i
t
h0ζSζ2ρ-1andsキ ρ，wed
I
fsi
se
v
e
n，t
h
e
n
Gs=
I
i+j
{{ い }
三
s
，i
幸 j(
mod 勾)}u{
{VS/
2，VS/2+P}}
I
fsi
soddand sキ ρ，t
h
e
n
叫
j
}I
i
:odd，i
-j三 s(mod
GS ={{ い'
Gsi
sa1
f
a
c
t
o
ro
fK2P andt
h
es
e
to
f Gs d
e
n
o
t
e
dby
GN
2ρ =
I
包 括 2ρ-い
吋)
{Gs
i
sap
e
r
f
e
c
t1
f
a
c
t
o
r
i
z
a
t
i
o
no
fK2P((
7
)
)
. Forexample
，GN
sshowni
nFIGURE3
.
lO i
8
8
GA2P andGl
¥
らP a
r
ei
s
o
m
o
r
p
h
i
cp
e
r
f
e
c
tl
f
a
c
t
o
r
i
z
a
t
i
o
n
s([
3
).
)
GN
lO
FIGURE3
4
.P
e
r
f
e
c
tl
f
a
c
t
o
r
i
z
a
t
i
o
no
fK16
Al
f
a
c
t
o
r
包a
t
i
o
nF i
sc
a
l
l
e
df
a
c
t
o
r
l
r
o
t
a
t
i
o
n
a
li
fF h
a
sanautomorphism f
i
x
i
n
g
twov
e
r
t
i
c
e
s(
a
n
donel
f
a
c
t
o
r
)，andp
e
r
m
u
t
i
n
gt
h
er
e
m
a
i
n
i
n
g 2n-2v
e
r
t
i
c
e
s(
a
n
d2n-2
l
f
a
c
t
o
r
s
)i
nas
i
n
g
l
ecy
c
1e
.I
t has a c
o
n
v
e
n
i
e
n
tg
e
o
m
e
t
r
i
cr
e
p
r
e
s
e
n
t
a
t
i
o
n
. One t
a
k
e
s
t
h
ev
e
r
t
i
c
e
so
ft
h
er
e
g
u
l
a
rp
o
l
y
g
o
nw
i
t
h2n-2 v
e
r
t
i
c
e
sandl
a
b
e
l
sthemw
i
t
he
l
e
m
e
n
t
s
h
eo
t
h
e
rtwov
e
r
t
i
c
e
si
sl
a
b
e
l
e
dw
i
t
h∞ 1，∞ 2，whereZ2n-2 d
e
n
o
t
e
st
h
er
e
s
o
f ~n-2; t
i
d
u
ec
l
a
s
sgroupmodulo2
n
2
.L
e
tF1 be a s
t
a
r
t
e
rl
f
a
c
t
o
ro
faf
a
c
t
o
r
l
r
o
t
a
t
i
o
n
a
l
l
f
a
c
t
o
r
i
z
a
t
i
o
nF
. The2n-2 l
f
a
c
t
o
r
sa
r
eo
b
t
a
i
n
e
dbyr
o
t
a
t
i
n
gt
h
ef
i
g
u
r
es
u
c
c
e
s
s
i
v
e
l
y
t
h
r
o
u
g
hana
n
g
l
e2
πj
(
2
n
2
)
.Fc
o
n
s
i
s
t
so
ft
h
e
s
e 2n-2 l
f
a
c
t
o
r
sandt
h
ef
i
x
e
dl
f
a
c
t
o
r
F
*
:
川i-j
F*={{i
叶u{{∞ 1，∞ 2}}
o=n-l(mod2n-
As
t
a
r
t
e
rl
f
a
c
t
o
ro
faf
a
c
t
o
r
l
r
o
t
a
t
i
o
n
a
l，p
e
r
f
e
c
tl
f
a
c
t
o
r
i
z
a
t
i
o
no
fK16 i
s shown i
n
FIGURE4
.
8
9
三シ
1
2
1
1
fli--114
.
/
6
00
2
FIGURE4
“
5
.P
e
r
f
e
c
tl
f
a
c
t
o
r
匂a
t
i
o
n
so
fK2
卸n f
o
r2n =28，
2
必
4
4，
鈍
3
4
4
L
e
tρbeap
r
i
m
enumberand m b
e
. an
a
t
u
r
a
lnumbers
u
c
hthatρm三 3(mod4
)
.
q-1)/
2and2n= q+1
. GF(q) d
e
n
o
t
e
st
h
eG
a
l
o
i
sf
i
e
l
dw
i
t
hq
Wep
u
t q =ρm，S=(
e
l
e
m
e
n
t
s
. K2n= (V
，E) d
e
n
o
t
e
st
h
e completegraphw
i
t
h2nv
e
r
t
i
c
e
s，and
{
∞
)
V= GF(q)U
L
e
tωbeap
r
i
m
i
t
i
v
ee
l
e
m
e
n
to
f GF(q). Wed
e
f
i
n
eas
t
a
r
t
e
r1
f
a
c
t
o
rFo:
九 ={{ω2iω川
I
i=川
2，"
'
， s-1}U {{い}}
ForanygεGF(q)，
Fg =Fo+g
={
{
ω 2i+ιJ1十 g}I
i=山，...， S
l
}
U
{
{
ι
∞
}
}
i
sa 1
-f
a
c
t
o
rwhichi
si
n
d
u
c
e
dbyt
h
es
t
a
r
t
e
rFo・ Then
I
gεGF
只附
(
ω
附
り
ω
q
)
i
詰
saト
1
一f
臼
a
c
t
or
i
包
z
副
a
t
i
∞
onぱ
0fι
K三
ふ
n. I
ti
s proved t
h
a
t F(ω
)i
ss
e
m
i
r
e
g
u
l
a
r(
(1
)
)
.Bys
u
i
t
a
b
l
e
F(ω)={
ι
ぽ
s
e
l
e
c
t
i
o
n
so
ft
h
es
e
m
i
r
e
g
u
l
a
r
s，wemayc
o
n
s
t
r
u
c
tp
e
r
f
e
c
t1
f
a
c
t
o
r
i
z
a
t
i
o
n
s
.
9
0
I
nc
a
s
eρ=3and m=3
，l
e
tωbe ap
r
i
m
i
t
i
v
ee
l
e
m
e
n
to
f GF(33) w
i
t
h a minimal
p
o
l
y
n
o
m
i
a
l x3 十 2X2+1
. ThenF(ω) i
sap
e
r
f
e
c
tl
f
a
c
t
o
r
i
z
a
t
i
o
no
f K28・
I
nc
a
s
eρ=3and m=5，l
e
t ωbe ap
r
i
m
i
t
i
v
ee
l
e
m
e
n
to
f GF(35) w
i
t
hanminimal
p
o
l
y
n
o
m
i
a
lX 5 +が +X2 +1
. ThenF(ω
5
)i
sap
e
r
f
e
c
t1
f
a
c
t
o
r
i
z
a
t
i
o
no
fι
4
4・
I
nc
a
s
eρ=7and m =3
，l
e
t ωbeap
r
i
m
i
t
i
v
ee
l
e
m
e
n
to
fGF(73) w
i
t
h a minimal
p
o
l
y
n
o
m
i
a
lX3+ぷ +X +2
. ThenF(ω
3
7
)i
sap
e
r
f
e
c
t1
f
a
c
t
o
r
i
z
a
t
i
o
no
fK344・
Acknowledgment. Thea
u
t
h
o
rwouldl
i
k
et
oe
x
p
r
e
s
sh
e
rt
h
a
n
k
st
oP
r
o
f
e
s
s
o
rZ
.
KiyasuandP
r
o
f
e
s
s
o
rG
.Nakamuraf
o
rt
h
e
i
rh
e
l
p
f
u
la
d
v
i
c
e
.
REFERENCES
(1
) B
.A
.Anderson，A c
l
a
s
so
fs
t
a
r
t
e
ri
n
d
u
c
e
d1
f
a
c
t
o
r
i
z
a
t
i
o
n
s，Gra
ρh
sand Com-
b
i
n
a
t
o
r
i
c
s
，L
e
c
t
u
r
eNotesi
nMathematics4
0
6，S
p
r
i
n
g
e
r
，NewYork(
1
9
7
4
)1
8
0
1
8
5
.
(2) B
.A.Anderson，Symmetryg
r
o
u
p
so
f somep
e
r
f
e
c
t1
f
a
c
t
o
r
i
z
a
t
i
o
n
so
fc
o
m
p
l
e
t
e
g
r
a
p
h
s
.D
i
s
c
r
e
t
eM
a
t
h
.1
8，2
2
7
2
3
4(
1
9
7
7
)
.
(3) M.Kobayashi，On p
e
r
f
e
c
t on~-factorization o
ft
h
ec
o
m
p
l
e
t
egraphK2P，t
oa
p
p
e
a
r
.
(4) M.Kobayashi，H.Awoki，Y
.NakazakiandG
.Nakamura，A p
e
r
f
e
c
to
n
e
f
a
c
t
o
r
i
z
a
t
i
o
no
fK3
oa
p
p
e
a
ri
nG
r
a
p
h
sandC
o
m
b
i
n
a
t
o
r
i
c
s
.
6，t
(5) M.KobayashiandK
i
y
a
s
u
Z
e
n
'
i
t
i，S
e
m
i
r
e
g
u
l
a
ro
n
e
f
a
c
t
o
r
i
z
a
t
i
o
n
so
ft
h
ec
o
m
p
l
e
t
e
ns
u
b
m
i
s
s
i
o
n
.
graphKpm+l，i
.MendelsohnandA
. Rosa，O
n
e
f
a
c
t
o
r
i
z
a
t
i
o
n
so
ft
h
ec
o
m
p
l
e
t
egraph- as
u
r
(6) E
ρh Theoη 9(
1
9
8
5
)4
3
6
5
.
vey，J Gra
(7) G
. Nakamura，D
u
d
n
e
y
'
sroundt
a
b
l
eproblemandt
h
ee
d
g
e
c
o
l
o
r
i
n
go
ft
h
ecomp
l
e
t
egraph(
i
nJ
a
p
a
n
e
s
e
)
.S
u
g
a
k
uSemina
γNo.1
5
9，2
4
2
9(
1
9
7
5
)
.
o
l
u
t
i
o
n
so
fD
u
d
e
n
e
y
'
s round t
a
b
l
e problem(
i
n
(8) G
. Nakamura and M.Tanaka，S
Japanese).RIMS K
o
k
y
u
r
o
k
u3
7
1(
1
9
7
9
)4
7
6
4
.