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 3pm to 3.50pm.

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 Travis Scrimshaw.