Decomposing oriented graphs into 6 locally irregular oriented graphs

An undirected graph G is locally irregular if every two of its adjacent vertices have distinct degrees. We say that G is decomposable into k locally irregular graphs if there exists a partition E1∪E2∪ ...∪Ek of the edge set E(G) such that each Ei induces a locally irregular graph. It was recently conjectured by Baudon et al. that every undirected graph admits a decomposition into 3 locally irregular graphs, except for a well-characterized set of indecomposable graphs. We herein consider an oriented version of this conjecture. Namely, can every oriented graph be decomposed into 3 locally irregular oriented graphs, i.e. whose adjacent vertices have distinct outdegrees? We start by supporting this conjecture by verifying it for several classes of oriented graphs. We then prove a weaker version of this conjecture. Namely, we prove that every oriented graph can be decomposed into 6 locally irregular oriented graphs. We finally prove that even if our conjecture were true, it would remain NP-complete to decide whether an oriented graph is decomposable into 2 locally irregular oriented graphs.