A complete solution to the generalized honeymoon Oberwolfach problem with one round table

Abstract

The generalized honeymoon Oberwolfach problem (HOP) asks whether it is possible to seat $2n$ participants consisting of $n$ newlywed couples at a conference with $s$ tables of size $2$ and $t$ "round'' tables of sizes $2m_1, 2m_2, \ldots, 2m_t$, where $n = s + \sum_{i=1}^{t} m_i $ with all $m_i \geq 2$, over several nights so that each participant sits next to their spouse every time and next to each other participant exactly once. We denote this problem by $HOP(2^{\langle s \rangle}, 2m_1, \ldots, 2m_t)$. In this paper, we provide a complete solution to the generalized HOP with one round table, showing that the obvious necessary conditions for $HOP(2^{\langle s \rangle}, 2m)$ to have a solution are also sufficient.