Given a 2-regular graph of order $v$, $F$ say, the Oberwolfach problem $OP(F)$, asks for a decomposition of the complete graph $K_{v}$ into copies of $F$. In this talk, we will discuss new results which show that $OP(F)$ has a solution whenever $F$ has a sufficiently large cycle which meets an explicit lower bound and in addition, has an involuntary automorphism acting as a reflection on exactly one of the cycles of $F$. We call such automorphisms single-flip automorphisms.

Our constructions use graceful labelings of 2-regular graphs with a vertex removed, which we call Zillion Graphs. We show that this class of graphs is graceful as long as the length of the path-component meets an explicit lower bound. A much better lower bound on the length of the path is given for an $\alpha$-labeling of such graphs to exist.

This is joint work with Andrea Burgess and Tommaso Traetta.

Bio: Peter Danziger is a professor in the Department of Mathematics at Toronto Metropolitan University (recently renamed) and has worked extensively on Design Theory, Graph Decompositions and Graph Factorizations.