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 \(\overrightarrow{G}\) can be assigned weights from {1, 2, 3} so that every two adjacent vertices of \(\overrightarrow{G}\) receive distinct sums of outgoing 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 $\mathsf{NP}$-complete problem. These results also hold for product or list versions of this problem.