Decomposition of Complete Bipartite Digraphs and Complete Digraphs into Directed Paths and Directed Cycles of Fixed Even Length

Abstract

In this paper, we give some necessary and sufficient conditions for decomposing the complete bipartite digraphs ${\mathcal D}K_{m, n}$ and complete digraphs ${\mathcal D}K_n$ into directed paths ${\mathop {P}\limits ^{\rightarrow }}_{k+1}$ and directed cycles ${\mathop {C}\limits ^{\rightarrow }}_{k}$ with $k$ arcs each. In particular, we prove that: (1) For any nonnegative integers $p$ and $q$; and any positive integers $m$, $n$, and $k$ with $m\ge k$ and $n\ge k$; a decomposition of ${\mathcal D} K_{m, n}$ into $p$ copies of ${\mathop {P}\limits ^{\rightarrow }}_{k+1}$ and $q$ copies of ${\mathop {C}\limits ^{\rightarrow }}_{k}$ exists if and only if $k(p+q)=2mn$, $p\ne 1$, $(m, n, k, p)\ne (2, 2, 2, 3)$, and $k$ is even when $q>0$. (2) For any nonnegative integers $p$ and $q$ and any positive integers $n$ and $k$ with $k$ even and $n\ge 2k$, a decomposition of ${\mathcal D} K_{n}$ into $p$ copies of ${\mathop {P}\limits ^{\rightarrow }}_{k+1}$ and $q$ copies of ${\mathop {C}\limits ^{\rightarrow }}_{k}$ exists if and only if $k(p+q)=n(n-1)$ and $p\ne 1$. We also give necessary and sufficient conditions for such decompositions to exist when $k=2$ or $4$.