An oriented version of the 1-2-3 Conjecture
Résumé
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 can be assigned weights from {1,2,3} so that every two adjacent vertices of ->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 NP-complete problem. These results also hold for product or list versions of this problem.
Origine : Fichiers produits par l'(les) auteur(s)
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