Existence of relative Heffter arrays for nontrivial subgroups

Determine the existence conditions for relative Heffter arrays over cyclic groups when the distinguished subgroup has order greater than one, thereby resolving the largely open existence problem beyond the currently known partial results.

Background

The paper distinguishes classical Heffter arrays, corresponding to the trivial subgroup, from relative Heffter arrays, in which the entries form a half-set outside a subgroup J of order t. The authors note that existence is completely established in the classical case: a square Heffter array H(n; k) exists exactly when n ≥ k ≥ 3.

For relative Heffter arrays with t > 1, the paper records that only partial existence results are known. This remains an unresolved foundational question for the broader theory of relative Heffter spaces and their associated graph decompositions and embeddings.