The Oberwolfach Problem ':' New Directions?

Mateja Šajna (University of Ottawa)

The celebrated Oberwolfach problem, over 50 years old and in general still open, asks whether \(n\) participants at a conference can be seated at \(k\) round tables of sizes \(m_1, \ldots, m_k\) (where \(m_1+ \ldots +m_k=n\)) so that over the course of several meals everybody sits next to everybody else exactly once. This problem can be modeled as a decomposition of the complete graph \(K_n\) into 2-factors, each consisting of \(k\) disjoint cycles of lengths \(m_1, \ldots, m_k\).

The Oberwolfach problem for tables of equal size was solved decades ago, and since then, solutions to many other special cases (for example, tables of even length, and exactly two tables) have been found. Then, in 2021, Glock, Joos, Kim, K"{u}hn, and Osthus published an impressive paper with the title ``Resolution of the Oberwolfach problem’’. But does that mean that the Oberwolfach problem has been completely solved?

This talk will serve as an introduction to the Oberwolfach problem and its variants, especially those that I have worked on in the past. I will then focus on my recent work on the directed version of the problem, wherein we are interested in decomposing \(K_{n}^\ast\), the complete symmetric digraph of order \(n\), into spanning subdigraphs, each a disjoint union of \(k\) directed cycles of lengths \(m_1, \ldots, m_k\) (where \(m_1+ \ldots +m_k=n\)). Such a decomposition models a seating arrangement of \(n\) participants at \(k\) tables of sizes \(m_1, \ldots, m_k\) such that everybody sits {\em to the right} of everybody else exactly once. I will present a recursive construction that generates solutions to many infinite families of cases of the directed Oberwolfach problem with variable cycle lengths, and discuss its potential to address other variants.

 Overview  Program