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

Abstract

The directed Oberwolfach problem OP${}^{\ast }(m_1,\dots ,m_k)$ asks whether the complete symmetric digraph $K_n^{\ast }$, assuming $n=m_1+\dots +m_k$, 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 OP${}^{\ast }(m_1,\dots ,m_k)$ given a solution to OP${}^{\ast }(m_1,\dots ,m_{\ell })$, for some $\ell <k$, if certain conditions on $m_1,\dots ,m_k$ are satisfied. This approach enables us to extend a solution for OP${}^{\ast }(m_1,\dots ,m_{\ell })$ into a solution for OP${}^{\ast }(m_1,\dots ,m_{\ell },t)$, as well as into a solution for OP${}^{\ast }(m_1,\dots ,m_{\ell },2^{⟨t⟩})$, 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 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 $2\le m_1+\dots +m_k\le 13$, OP${}^{\ast }(m_1,\dots ,m_k)$ has a solution if and only if $(m_1,\dots ,m_k)\notin \{(4),(6),(3,3)\}$.