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 − → uv, 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.

A graph $G$ is said to be $(k,m)$-choosable if for any assignment of $k$-element lists $L_v \subset \mathbb{R}$ to the vertices $v \in V(G)$ and any assignment of $m$-element lists $L_e \subset \mathbb{R}$ to the edges $e \in E(G)$  there exists a total weighting $w: V(G) \cup E(G) \rightarrow \mathbb{R}$ of $G$ such that $w(v) \in L_v$ for any vertex $v \in V(G)$ and $w(e) \in L_e$ for any edge $e \in E(G)$ and furthermore, such that for any pair of adjacent vertices $u,v$, we have $w(u)+ \sum_{e \in E(u)}w(e) \neq w(v)+ \sum_{e \in E(v)}w(e)$, where $E(u)$ and $E(v)$ denote the edges incident to $u$ and $v$ respectively. In this paper we give an algorithmic proof showing that any graph $G$ without isolated edges is $(1, 2 \lceil \log_2(\Delta(G)) \rceil+1)$-choosable, where $\Delta(G)$ denotes the maximum degree in $G$.

For any S ⊂ Z we say that a graph G has the S-property if there exists an S-edge-weighting w : E(G) → S such that for any pair of adjacent vertices u, v we have e∈E(v) w(e) = e∈E(u) w(e), where E(v) and E(u) are the sets of edges incident to v and u, respectively. This work focuses on {a, a + 2}-edge-weightings where a ∈ Z is odd. We show that a 2-connected bipartite graph has the {a, a + 2}-property if and only if it is not a socalled odd multi-cactus. In the case of trees, we show that only one case is pathological. That is, we show that all trees have the {a, a + 2}-property for odd a = −1, while there is an easy characterization of trees without the {−1, 1}-property.

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We prove that every connected graph can be edge-decomposed into a spanning tree, an even graph, and a star forest.