On the Generalized Honeymoon Oberwolfach Problem

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 $$\mathrm{HOP}(2^{\langle s \rangle}, 2m_1, \ldots, 2m_t)$$. This paper is the first of two papers investigating the generalized HOP. While the second paper will deal with the generalized HOP with a single round table (i.e. table of size at least $4$), the present work develops solutions for the generalized HOP with multiple round tables. In particular, we present solutions to certain cases with two round tables, showing that a solution to $$\mathrm{HOP}(2^{\langle s \rangle}, 2m_1, 2m_2)$$ exists when $n \equiv 1 \pmod{(2m_1 + 2m_2)}$ or $n \equiv m_1 + m_2 \pmod{(2m_1 + 2m_2)}$. We also develop solutions for cases with small round tables, showing that $$\mathrm{HOP}(2^{\langle s \rangle}, 2m_1, \dots, 2m_t)$$ has a solution whenever $m = m_1 + \dots + m_t \leq 10$, $n = s + m$ is odd, and $n(n - 1) \equiv 0 \pmod{2m}$.