Bounded degree acyclic decompositions of digraphs

An acyclic decomposition of a digraph is a partition of the edges into acyclic subgraphs. Trivially every digraph has an acyclic decomposition into two subgraphs. It is proved that for every integer s⩾2 every digraph has an acyclic decomposition into s subgraphs such that in each subgraph the outdegree of each vertex v is at most deg(v)s−1. For all digraphs this degree bound is optimal.