Plenary: The Directed Oberwolfach Problem

The Oberwolfach Problem, originally posed by Ringel in 1967, asks whether n people can be seated at round tables of sizes k1, …, kt, where k1 + ⋯ + kt=n, over successive nights in such a way that each person sits next to each other person exactly once. In other words, does the complete graph Kn admit a 2-factorization in which each 2-factor consists of cycles of lengths k1, …, kt? Over the course of the next six decades, many instances of the Oberwolfach Problem have been solved, including for uniform 2-factors, 2-factors containing exactly two components and bipartite 2-factors. While an asymptotic existence result is known, a complete solution to the Oberwolfach problem remains elusive.

The Oberwolfach Problem has also been extended to the directed case, where we seek to find a 2-factorization of the complete symmetric digraph Kn*. Echoing some of the history of the original problem, instances for which the directed Oberwolfach Problem has been settled include uniform directed 2-factors and directed 2-factors with exactly two components.

In this talk, we give an overview of the history of the Oberwolfach Problem and its directed variant. We then discuss a recent result which completely solves the directed Oberwolfach problem with bipartite 2-factors when the order n is congruent to 2 modulo 4. This is joint work with Peter Danziger and Alice Lacaze-Masmonteil.