Bailey–Stevens conjecture on tight Hamiltonian-cycle decompositions

Determine whether complete k-uniform hypergraphs admit decompositions into tight Hamiltonian cycles in the case corresponding to the coprime case of Baranyai’s wreath conjecture.

Background

The paper identifies the case $\gcd(n,k)=1$ of Baranyai’s wreath conjecture as especially difficult and relates it to a corresponding case of the Bailey–Stevens conjecture. That conjecture concerns decompositions of complete k-uniform hypergraphs into tight Hamiltonian cycles.

The authors explicitly report that both conjectures remain widely open despite work by multiple researchers. This unresolved status motivates the paper’s algebraic study of the wreath matrix, although the matrix results do not settle either decomposition conjecture.