On the spouse-loving variant of the Oberwolfach problem

TL;DR

This paper investigates the decomposition of a modified complete graph into 2-factors with cycles of specified lengths, extending the understanding of the Oberwolfach problem with new existence conditions.

Contribution

It provides necessary and sufficient conditions for decomposing $K_n+I$ into 2-factors with cycles of certain lengths, including cases with all even cycle lengths.

Findings

Decomposition exists if and only if cycle length divides n, with a specific exception.

Decomposition into 2-factors with all even cycle lengths is always possible.

Conditions extend the classical Oberwolfach problem to a modified graph setting.

Abstract

We prove that $K_n+I$, the complete graph of an even order with a $1$-factor duplicated, admits a decomposition into $2$-factors, each a disjoint union of cycles of length $m \geq 5$ if and only if $m \mid n$, except possibly when $m$ is odd and $n=4m$. In addition, we show that $K_n+I$ admits a decomposition into $2$-factors, each a disjoint union of cycles of lengths $m_1, \ldots, m_t$, whenever $m_1, \ldots, m_t$ are all even.