The {\em Oberwolfach Problem} was posed in the 1960s by Ringel as a seating problem: $v$ people attend a conference in Oberwolfach, where the dining room has round tables of sizes $m_1, m_2, \ldots, m_t$ (with $m_1 + \cdots + m_t = v$). Is it possible, over $(v-1)/2$ successive nights, to devise a seating plan in which each person sits next to each other person exactly once?

In other words, the Oberwolfach Problem asks whether, given a 2-factor $F$ of order $v$, the complete graph $K_v$ admits a 2-factorization with each 2-factor isomorphic to $F$. It is easy to see that such a factorization can exist only if $v$ is odd. For even $v$, it is common to instead decompose $K_v-I$, the complete graph with the edges of a 1-factor removed; this is sometimes called the {\em spouse-avoiding variant}.

In this talk, we present recent results on the Oberwolfach Problem obtained via graceful labellings. We then discuss several natural variants of the problem, namely the {\em spouse-loving variant} (in which we factor the complete graph with the edges of a 1-factor added) and the {\em directed Oberwolfach Problem}.

This talk contains joint work with Noah Bolohan, Iona Buchanan, Peter Danziger, Nevena Franceti\'{c}, Mateja \v{S}ajna, Tommaso Traetta and Ryan Van Snick.

### About Pure mathematics seminars

We present regular seminars on a range of pure mathematics interests. Students, staff and visitors to UQ are welcome to attend, and to suggest speakers and topics.

Seminars are usually held on Tuesdays from 2 to 3pm.

Talks comprise 45 minutes of speaking time plus five minutes for questions and discussion.

#### Information for speakers

Researchers in all pure mathematics fields attend our seminars, so please aim your presentation at a general mathematical audience.

#### Contact us

To volunteer to talk or to suggest a speaker, email Ole Warnaar or Ramiro Lafuente.