On Proper Labellings of Graphs with Minimum Label Sum

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