Decomposing oriented graphs into 6 locally irregular oriented graphs

Julien Bensmail (1) , Gabriel Renault (1)

1 LaBRI - Laboratoire Bordelais de Recherche en Informatique

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 E_1 \cup E_2 \cup ... \cup E_k of the edge set E(G) such that each E_i 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.

Domaines: Mathématique discrète [cs.DM]

Julien Bensmail, Gabriel Renault. Decomposing oriented graphs into 6 locally irregular oriented graphs. 2014. ⟨hal-00869778v2⟩