AN ORIENTED VERSION OF THE 1-2-3 CONJECTURE

The well-known 1-2-3 Conjecture addressed by Karonski, 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) over right arrow can be assigned weights from {1, 2, 3} so that every two adjacent vertices of (G) over right arrow 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 NP-complete problem. These results also hold for product or list versions of this problem.