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The directed Oberwolfach problem with variable cycle lengths: a recursive construction
Authors
Suzan Kadri
Mateja Ε ajna
Publication date
21 September 2023
Publisher
View
on
arXiv
Abstract
The directed Oberwolfach problem OP
β
(
m
1
,
β¦
,
m
k
)
^\ast(m_1,\ldots,m_k)
β
(
m
1
β
,
β¦
,
m
k
β
)
asks whether the complete symmetric digraph
K
n
β
K_n^\ast
K
n
β
β
, assuming
n
=
m
1
+
β¦
+
m
k
n=m_1+\ldots +m_k
n
=
m
1
β
+
β¦
+
m
k
β
, admits a decomposition into spanning subdigraphs, each a disjoint union of
k
k
k
directed cycles of lengths
m
1
,
β¦
,
m
k
m_1,\ldots,m_k
m
1
β
,
β¦
,
m
k
β
. We hereby describe a method for constructing a solution to OP
β
(
m
1
,
β¦
,
m
k
)
^\ast(m_1,\ldots,m_k)
β
(
m
1
β
,
β¦
,
m
k
β
)
given a solution to OP
β
(
m
1
,
β¦
,
m
β
)
^\ast(m_1,\ldots,m_\ell)
β
(
m
1
β
,
β¦
,
m
β
β
)
, for some
β
<
k
\ell<k
β
<
k
, if certain conditions on
m
1
,
β¦
,
m
k
m_1,\ldots,m_k
m
1
β
,
β¦
,
m
k
β
are satisfied. This approach enables us to extend a solution for OP
β
(
m
1
,
β¦
,
m
β
)
^\ast(m_1,\ldots,m_\ell)
β
(
m
1
β
,
β¦
,
m
β
β
)
into a solution for OP
β
(
m
1
,
β¦
,
m
β
,
t
)
^\ast(m_1,\ldots,m_\ell,t)
β
(
m
1
β
,
β¦
,
m
β
β
,
t
)
, as well as into a solution for OP
β
(
m
1
,
β¦
,
m
β
,
2
β¨
t
β©
)
^\ast(m_1,\ldots,m_\ell,2^{\langle t \rangle})
β
(
m
1
β
,
β¦
,
m
β
β
,
2
β¨
t
β©
)
, where
2
β¨
t
β©
2^{\langle t \rangle}
2
β¨
t
β©
denotes
t
t
t
copies of 2, provided
t
t
t
is sufficiently large. In particular, our recursive construction allows us to effectively address the two-table directed Oberwolfach problem. We show that OP
β
(
m
1
,
m
2
)
^\ast(m_1,m_2)
β
(
m
1
β
,
m
2
β
)
has a solution for all
2
β€
m
1
β€
m
2
2 \le m_1\le m_2
2
β€
m
1
β
β€
m
2
β
, with a definite exception of
m
1
=
m
2
=
3
m_1=m_2=3
m
1
β
=
m
2
β
=
3
and a possible exception in the case that
m
1
β
{
4
,
6
}
m_1 \in \{ 4,6 \}
m
1
β
β
{
4
,
6
}
,
m
2
m_2
m
2
β
is even, and
m
1
+
m
2
β₯
14
m_1+m_2 \ge 14
m
1
β
+
m
2
β
β₯
14
. It has been shown previously that OP
β
(
m
1
,
m
2
)
^\ast(m_1,m_2)
β
(
m
1
β
,
m
2
β
)
has a solution if
m
1
+
m
2
m_1+m_2
m
1
β
+
m
2
β
is odd, and that OP
β
(
m
,
m
)
^\ast(m,m)
β
(
m
,
m
)
has a solution if and only if
m
β
3
m \ne 3
m
ξ
=
3
. In addition to solving many other cases of OP
β
^\ast
β
, we show that when
2
β€
m
1
+
β¦
+
m
k
β€
13
2 \le m_1+\ldots +m_k \le 13
2
β€
m
1
β
+
β¦
+
m
k
β
β€
13
, OP
β
(
m
1
,
β¦
,
m
k
)
^\ast(m_1,\ldots,m_k)
β
(
m
1
β
,
β¦
,
m
k
β
)
has a solution if and only if
(
m
1
,
β¦
,
m
k
)
βΜΈ
{
(
4
)
,
(
6
)
,
(
3
,
3
)
}
(m_1,\ldots,m_k) \not\in \{ (4),(6),(3,3) \}
(
m
1
β
,
β¦
,
m
k
β
)
ξ
β
{(
4
)
,
(
6
)
,
(
3
,
3
)}
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Last time updated on 12/10/2023