An oriented version of the 1-2-3 Conjecture

Olivier Baudon (1) , Julien Bensmail (1) , Eric Sopena (1)

1 LaBRI - Laboratoire Bordelais de Recherche en Informatique

The well-known 1-2-3 Conjecture addressed by Karo nski, Luczak and Thomason asks whether the edges of every undirected graph G with no isolated edge can be assigned weights from {1,2,3} so that the sum of incident weights at each vertex yields a proper vertex-colouring of G. In this work, we consider a similar problem for oriented graphs. We show that the arcs of every oriented graph G can be assigned weights from {1,2,3} so that every two adjacent vertices of G receive distinct sums of outgoing (or ingoing) weights. This result is tight in the sense that some oriented graphs do not admit such an assignment using the weights from {1,2} only. We fi nally prove that deciding whether two weights are suffi cient for a given oriented graph is an NP-complete problem. These results also hold for a product version of the problem.

Mots clés: oriented graph; neighbour-sum-distinguishing arc-weighting; complexity; 1-2-3 Conjecture

Domaines: Mathématique discrète [cs.DM]

Olivier Baudon, Julien Bensmail, Eric Sopena. An oriented version of the 1-2-3 Conjecture. 2013. ⟨hal-00849065v2⟩