We prove that the product version of the 1-2-3 Conjecture, raised by Skowronek-Kaziów in 2012, is true. Namely, for every connected graph with order at least 3, we prove that we can assign labels 1, 2, 3 to the edges in such a way that no two adjacent vertices are incident to the same product of labels.
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Recall that a proper k-vertex-colouring of a graph G is a partition \((V_1, \dots , V_k\)) of V(G) where all \(V_i\)’s are independent. The chromatic number \(\chi (G)\) of G is the smallest \(k \ge 1\) such that proper k-vertex-colourings of G exist. We say that G is k-colourable if \(\chi (G) \le k\).
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The authors are grateful to the three anonymous referees for their careful reading of a previous version of the current work, which allowed to improve the general quality and correctness not only of the main proof, but also of the whole paper.
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Bensmail, J., Hocquard, H., Lajou, D. et al. A Proof of the Multiplicative 1-2-3 Conjecture. Combinatorica 43, 37–55 (2023). https://doi.org/10.1007/s00493-023-00003-0