1-2-3 Conjecture in digraphs: More results and directions

Horňak, Przybyło and Woźniak recently proved that, a small class of obvious exceptions apart, every digraph can be 4-arc-weighted so that, for every arc uv⃗, the sum of weights incoming to u is different from the sum of weights outgoing from v. They conjectured a stronger result, namely that the same statement with 3 instead of 4 should also be true. We verify this conjecture in this work. This work takes place in a recent “quest” towards a directed version of the 1–2–3 Conjecture, the variant above being one of the last introduced ones. We take the occasion of this work to establish a summary of all results known in this field, covering known upper bounds, complexity aspects, and choosability. On the way we prove additional results which were missing in the whole picture. We also mention the aspects that remain open.