The directed Oberwolfach problem with variable cycle lengths: a recursive construction

The directed Oberwolfach problem OP$^\ast(m_1,\ldots,m_k)$ asks whether the complete symmetric digraph $K_n^\ast$, assuming $n=m_1+\ldots +m_k$, admits a decomposition into spanning subdigraphs, each a disjoint union of $k$ directed cycles of lengths $m_1,\ldots,m_k$. We hereby describe a method for constructing a solution to OP$^\ast(m_1,\ldots,m_k)$ given a solution to OP$^\ast(m_1,\ldots,m_\ell)$, for some $\ell<k$, if certain conditions on $m_1,\ldots,m_k$ are satisfied. This approach enables us to extend a solution for OP$^\ast(m_1,\ldots,m_\ell)$ into a solution for OP$^\ast(m_1,\ldots,m_\ell,t)$, as well as into a solution for OP$^\ast(m_1,\ldots,m_\ell,2^{\langle t \rangle})$, where $2^{\langle t \rangle}$ denotes $t$ copies of 2, provided $t$ is sufficiently large. In particular, our recursive construction allows us to effectively address the two-table directed Oberwolfach problem. We show that OP$^\ast(m_1,m_2)$ has a solution for all $2 \le m_1\le m_2$, with a definite exception of $m_1=m_2=3$ and a possible exception in the case that $m_1 \in \{ 4,6 \}$, $m_2$ is even, and $m_1+m_2 \ge 14$. It has been shown previously that OP$^\ast(m_1,m_2)$ has a solution if $m_1+m_2$ is odd, and that OP$^\ast(m,m)$ has a solution if and only if $m \ne 3$. In addition to solving many other cases of OP$^\ast$, we show that when $2 \le m_1+\ldots +m_k \le 13$, OP$^\ast(m_1,\ldots,m_k)$ has a solution if and only if $(m_1,\ldots,m_k) \not\in \{ (4),(6),(3,3) \}$