The Oberwolfach Problem and its variants

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.