Recent advances on the directed Oberwolfach problem

A directed variant of the famous Oberwolfach problem, the directed Oberwolfach problem considers the following scenario. Given $n$ people seated at $t$ round tables of size $m_1,m_2,\ldots,m_t$, respectively, such that $m_1+m_2+\cdots +m_t=n$, does there exist a set of $n−1$ seating arrangements such that each person is seated to the right of every other person precisely once? I will first demonstrate how this problem can be formulated as a type of graph-theoretic problem known as a cycle decomposition problem. Then, I will discuss a particular style of construction that was first introduced by R. HÄggkvist in 1985 to solve several cases of the original Oberwolfach problem. Lastly, I will show how this approach can be adapted to the directed Oberwolfach problem, thereby allowing us to obtain solutions for previously open cases. Results discussed in this talk arose from collaborations with Andrea Burgess, Peter Danziger, and Daniel Horsley.