8 problems

Approximate Oberwolfach Nash-Williams' conjecture. For every $\varepsilon>0$, for all sufficiently large $n$, if

Let $m$ be a positive odd integer. The spouse-loving variant of the Oberwolfach problem, denoted $\operatorname{OP}^+(n;m)$, asks whether the graph $K_n+I$ admits a $C_m$-factoriza…

Let $n=\frac{1}{2}(m_1+\cdots+m_t)$, and let $\operatorname{HOP}(m_1,\ldots,m_t)$ denote the problem of decomposing $K_{2n}+(2n-3)I$ into 2-factors that are vertex-disjoint unions…

Let $k\geq 3$, and let $n$ be sufficiently large in terms of $k$ with $k\mid\binom{n-1}{k-1}$. A tight cycle in a $k$-graph is a cycle whose vertices can be cyclically ordered so t…

Bailey–Stevens conjecture. The complete hypergraph $K_n^k$ has a decomposition into tight Hamilton cycles if and only if

Minimum-degree Oberwolfach conjecture. For every $\varepsilon>0$, the following holds for sufficiently large $n$: if

Buratti–Sajna conjecture. $K_{2m}^*$ admits a resolvable directed $m$-cycle decomposition if and only if $m\geq 5$.

For a positive integer $v$ and a partition $\pi$ of $v$, let $OP(v;\pi)$ ask for a two-factorization of the complete graph (or of the complete graph minus a $1$-factor when appropr…