On the directed Oberwolfach problem for complete symmetric equipartite digraphs and uniform-length cycles

Abstract

We examine the necessary and sufficient conditions for a complete symmetric equipartite digraph $K_{n[m]}^{\ast }$ with $n$ parts of size $m$ to admit a resolvable decomposition into directed cycles of length $t$. We show that the obvious necessary conditions are sufficient for $m,n,t\ge 2$ in each of the following four cases: (i) $m(n-1)$ is even; (ii) $\gcd (m,n)\notin \{1,3\}$; (iii) $\gcd (m,n)=1$ and $4|n$ or $6|n$; and (iv) $\gcd (m,n)=3$, and if $n=6$, then $p|m$ for a prime $p\le 37$.