On Proper Labellings of Graphs with Minimum Label Sum

The 1-2-3 Conjecture states that every nice graph G (without component isomorphic to [Formula: see text]) 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 is that every nice 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 edge by at most three parallel edges. In other words, for every nice graph G, there should exist a locally irregular multigraph M with the same adjacencies and having few edges. We study proper labellings of graphs with the extra requirement that the sum of assigned labels must be as small as possible. That is, given a graph G, we are looking for a locally irregular multigraph [Formula: see text] with the fewest edges possible that can be obtained from G by replacing edges with parallel edges. This problem is quite different from the 1-2-3 Conjecture, as we prove that there is no k such that [Formula: see text] can always be obtained from G by replacing each edge with at most k parallel edges. We investigate several aspects of this problem. We prove that the problem of designing proper labellings with minimum label sum is [Formula: see text]-hard in general, but solvable in polynomial time for graphs with bounded treewidth. We also conjecture that every nice connected graph G admits a proper labelling with label sum at most [Formula: see text], which we verify for several classes of graphs.