An oriented version of the 1-2-3 Conjecture

The well-known 1-2-3 Conjecture addressed by Karoński, 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 finally prove that deciding whether two weights are sufficient for a given oriented graph is an NPcomplete problem. These results also hold for a product version of the problem.