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 −G⃗ can be assigned weights from {1, 2, 3} so that every two adjacent vertices of −G⃗ 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.
@article{bwmeta1.element.doi-10_7151_dmgt_1791,
author = {Olivier Baudon and Julien Bensmail and \'Eric Sopena},
title = {An Oriented Version of the 1-2-3 Conjecture},
journal = {Discussiones Mathematicae Graph Theory},
volume = {35},
year = {2015},
pages = {141-156},
zbl = {1326.05043},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.doi-10_7151_dmgt_1791}
}
Olivier Baudon; Julien Bensmail; Éric Sopena. An Oriented Version of the 1-2-3 Conjecture. Discussiones Mathematicae Graph Theory, Tome 35 (2015) pp. 141-156. http://gdmltest.u-ga.fr/item/bwmeta1.element.doi-10_7151_dmgt_1791/
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