The Directed Oberwolfach Problem With Variable Cycle Lengths: A Recursive Construction

The directed Oberwolfach problem ${\text{OP}}^{*}\left(m_1,\dots ,m_k\right)$ asks whether the complete symmetric digraph $K_n^{*}$, assuming $n=m_1+\cdots +m_k$, admits a decomposition into spanning subdigraphs, each a disjoint union of $k$ directed cycles of lengths $m_1,\dots ,m_k$. We hereby describe a method for constructing a solution to ${\text{OP}}^{*}\left(m_1,\dots ,m_k\right)$ given a solution to ${\text{OP}}^{*}\left(m_1,\dots ,m_{\ell }\right)$, for some $\ell <k$, if certain conditions on $m_1,\dots ,m_k$ are satisfied. This approach enables us to extend a solution for ${\text{OP}}^{*}\left(m_1,\dots ,m_{\ell }\right)$ into a solution for ${\text{OP}}^{*}\left(m_1,\dots ,m_{\ell },t\right)$, as well as into a solution for ${\text{OP}}^{*}\left(m_1,\dots ,m_{\ell },2^{〈t〉}\right)$, where $2^{〈t〉}$ 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 ${\text{OP}}^{*}\left(m_1,m_2\right)$ 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 $m_1\in \left\{4,6\right\}$, $m_2$ is even, and $m_1+m_2\ge 14$. It has been shown previously that ${\text{OP}}^{*}\left(m_1,m_2\right)$ has a solution if $m_1+m_2$ is odd, and that ${\text{OP}}^{*}\left(m,m\right)$ has a solution if and only if $m\ne 3$. In addition to solving many other cases of ${\text{OP}}^{*}$, we show that when $2\le m_1+\cdots +m_k\le 13$, ${\text{OP}}^{*}\left(m_1,\dots ,m_k\right)$ has a solution if and only if $\left(m_1,\dots ,m_k\right)\notin \left\{\left(4\right),\left(6\right),\left(3,3\right)\right\}$.