Horňak, Przybylo and Woźniak recently proved that, a small class of obvious exceptions apart, every digraph can be 4-arc-weighted so that, for every arc u->v, 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.

中文翻译：

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1-2-3 有向图中的猜想：更多结果和方向

Horňak、Przybylo 和 Woźniak 最近证明，除了一小类明显的例外，每个有向图都可以是 4 弧加权的，因此，对于每个弧 u->v，传入 u 的权重之和不同于从 v 传出的权重。他们推测了一个更强的结果，即相同的陈述 3 而不是 4 也应该是真的。我们在这项工作中验证了这个猜想。这项工作发生在最近对 1-2-3 猜想的定向版本的“探索”中，上面的变体是最后引入的变体之一。我们利用这项工作来总结该领域已知的所有结果，涵盖已知的上限、复杂性方面和可选择性。在此过程中，我们证明了整体情况中缺少的其他结果。我们还提到了仍然开放的方面。