Cyclic \((0,1)\)-Factorizations of The Complete Graph

Hartman and Rosa have shown that the complete graph \(K_{2n}\) has an acyclic one-factorization if and only if \(n\) is not a power of \(2\) exceeding \(2\). Here, we consider the following problem: for which \(n > 0\) and \(0 < k < \frac{n}{2}\) does the complete graph \(K_n\) admit a cyclic decomposition into matchings of size \(k\)? We give a complete solution to this problem and apply it to obtain a new class of perfect coverings.