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

Abstract

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.