A complete solution to the generalized HOP with one large table

Masoomeh Akbari (University of Ottawa)

The Honeymoon Oberwolfach Problem (HOP), introduced by \v{S}ajna, is a recent variant of the classic Oberwolfach Problem. This problem asks whether it is possible to seat \(2m_1 + 2m_2 + \dots + 2m_t = 2n\) participants, consisting of \(n\) newlywed couples, at \(t\) round tables of sizes \(2m_1, 2m_2, \dots, 2m_t\) for \(2n - 2\) successive nights, so that each participant sits next to their spouse every night and next to every other participant exactly once. This problem is denoted by \(\mathrm{HOP}(2m_1, 2m_2, \ldots, 2m_t)\). Jerade, Lepine, and \v{S}ajna have studied the HOP and resolved several important cases.

We generalized the HOP by allowing tables of size two, relaxing the previous restriction that tables must have a minimum size of four. In the generalized HOP, we aim to seat the \(2n\) participants at \(s\) tables of size \(2\) and \(t\) round tables of sizes \(2m_1, 2m_2, \dots, 2m_t\), where \(2n = 2s + 2m_1 + 2m_2 + \dots + 2m_t\) and \(m_i \geq 2\), while preserving the adjacency conditions of the HOP. We denote this problem by \(\mathrm{HOP}(2^{\langle s \rangle}, 2m_1, \ldots, 2m_t)\).

In this talk, we present a general approach to this problem and provide a solution to the generalized HOP with a single large table, showing that the necessary condition for \(\mathrm{HOP}(2^{\langle s \rangle}, 2m)\) to have a solution is also sufficient.

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