On Proper Labellings of Graphs with Minimum Label Sum

The 1–2–3 Conjecture, raised by Karoński, Łuczak and Thomason, states that almost every graph G admits a proper 3-labelling, i.e., labelling of the edges with 1, 2, 3 such no two adjacent vertices are incident to same sum labels. Another interpretation this conjecture, may be attributed Chartrand et al., is can turned into locally irregular multigraph M, having degree, replacing each its at most three parallel edges. In other words, for there should M adjacencies relatively small number if true, would indeed imply an $$|E(M)| \le 3|E(G)|$$ . work, we study labellings graphs extra requirement assigned labels must as possible. given G, looking $$M^*$$ smallest possible obtained from multiplying This problem actually quite different prove absolute constant k always edge We investigate several aspects problem, covering algorithmic combinatorial aspects. particular, designing minimum label $${\mathcal {N}}{\mathcal {P}}$$ -hard in general, but solvable polynomial time bounded treewidth. also conjecture connected 2|E(G)|, which verify classes graphs.