The Honeymoon Oberwolfach Problem is a surprisingly interesting variation on the spouse-avoiding variant of the Oberwolfach Problem. As a scheduling problem, $HOP(2m_{1}, ..., 2m_{t})$ asks whether it is possible to arrange $n = m_{1}+...+m_{t}$ couples at a conference at $t$ round tables of sizes $2m_{1}, ..., 2m_{t}$ for $2n - 2$ meals so that each participant sits next to their spouse at every meal, and sits next to every other participant exactly once. In graph-theoretic terms, a solution to $HOP(2m_{1}, ..., 2m_{t})$ is a decomposition of $K_{2n}+(2n-3)I$, the complete graph on $2n$ vertices with $2n-3$ additional copies of a chosen 1-factor $I$, into 2-factors, each consisting of disjoint $I$-alternating cycles of lengths $2m_{1}, ..., 2m_{t}$. It is also equivalent to a semi-uniform 1-factorization of $K_{2n}$ of type $(2m_{1}, ..., 2m_{t})$. Thus, the Honeymoon Oberwolfach Problem relates to Doug Stinson's work on the Oberwolfach Problem (1987, 1989), as well as on sequentially uniform 1-factorizations (2005).

I will present several results, most notably, a complete solution to the case with uniform cycle lengths. This is joint work with my students Dene Lepine and Mary Rose Jerade.

Bio: Dr. Mateja Šajna is a Professor in the Department of Mathematics and Statistics, University of Ottawa. She received her BMath in Applied Mathematics from the University of Ljubljana in her native Slovenia, and her MSc and Ph.D. from Simon Fraser University. In 2003, she was awarded a Kirkman Medal by the Institute of Combinatorics and its Applications, primarily for her Ph.D. work on cycle decompositions of complete graphs. This area of research remains her central focus, with a recent emphasis on variants of the Oberwolfach problem. Her interests include symmetry in graphs, eulerian properties of hypergraphs, and interdisciplinary applications of graphs.