The Honeymoon Oberwolfach Problem: small cases

TL;DR

This paper extends the verification of the Honeymoon Oberwolfach Problem solutions for cases with up to 20 couples using computational methods, building on previous results for smaller cases.

Contribution

The authors computationally verify the Honeymoon Oberwolfach Problem for all cases with up to 20 couples, expanding known solutions beyond previous smaller cases.

Findings

Confirmed solutions for all cases with n ≤ 20 couples.

Extended previous results from n ≤ 9 to n ≤ 20.

Supported the conjecture that solutions exist under necessary conditions.

Abstract

The Honeymoon Oberwolfach Problem HOP$(2m_1,2m_2,\ldots,2m_t)$ asks the following question. Given $n=m_1+m_2+\ldots +m_t$ newlywed couples at a conference and $t$ round tables of sizes $2m_1,2m_2,\ldots,2m_t$, is it possible to arrange the $2n$ participants at these tables 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? A solution to HOP$(2m_1,2m_2,\ldots,2m_t)$ is a decomposition of $K_{2n}+(2n-3)I$, the complete graph $K_{2n}$ with $2n-3$ additional copies of a fixed 1-factor $I$, into 2-factors, each consisting of disjoint $I$-alternating cycles of lengths $2m_1,2m_2,\ldots,2m_t$. The Honeymoon Oberwolfach Problem was introduced in a 2019 paper by Lepine and \v{S}ajna. The authors conjectured that HOP$(2m_1,2m_2,\ldots,$ $2m_t)$ has a solution whenever the obvious necessary conditions are satisfied, and proved the conjecture for several large cases, including the uniform cycle length case $m_1=\ldots=m_t$, and the small cases with $n \le 9$. In the present paper, we extend the latter result to all cases with $n \le 20$ using a computer search.