Geodesic
On {a, b}-Edge-Weightings of Bipartite Graphs with Odd a, b
Discussiones Mathematicae. Graph Theory, Tome 42 (2022) no. 1, pp. 159-185

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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 so-called 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.
Keywords: neighbour-sum-distinguishing edge-weightings, bipartite graphs, odd weights, 1-2-3 Conjecture 
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Bensmail, Julien; Inerney, Fionn Mc; Lyngsie, Kasper Szabo. On {a, b}-Edge-Weightings of Bipartite Graphs with Odd a, b. Discussiones Mathematicae. Graph Theory, Tome 42 (2022) no. 1, pp. 159-185. http://geodesic.mathdoc.fr/item/DMGT_2022_42_1_a10/