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 $n(n - 1) \equiv 0 \pmod{2m}$.

Figures (5)

• Figure 1: The $F_i$ subgraph in Lemma \ref{['lem:Newtool-1b']} for $m\equiv 2\ ({\rm mod}\ 4)$.
• Figure 2: Subgraph $F_i$ in Lemma \ref{['lem:Newtool-T2']} for $m\equiv 1\ ({\rm mod}\ 4)$ and $m\geq 9$.
• Figure 3: $(C_3, C_3)$-subgraphs of $4K_{9}^{\bullet}$ that generate an HOP $(C_3,C_3)$-decomposition for $4K_{9}^{\bullet}$.
• Figure 4: $(C_2, C_4, C_4)$-subgraphs of $2K_{25}^{\circ}$ that generate an HOP $(C_2, C_4, C_4)$-decomposition of $4K_{25}^{\bullet}$.
• Figure 5: $(C_2, C_3, C_5)$-subgraphs of $4K_{25}^{\bullet}$ that generate an HOP $(C_2, C_3, C_5)$-decomposition of $4K_{25}^{\bullet}$.

Theorems & Definitions (38)

• Theorem 1.1
• Theorem 1.2
• Definition 3.1
• Definition 3.2
• Theorem 3.3
• Lemma 3.4
• Lemma 3.5
• Lemma 3.6
• Lemma 3.7
• Theorem 3.8