An injective version of the 1-2-3 Conjecture

In this work, we introduce and study a new graph labelling problem standing as a combination of the 1-2-3 Conjecture and injective colouring of graphs, which also finds connections with the notion of graph irregularity. In this problem, the goal, given a graph G, is to label the edges of G so that every two vertices having a common neighbour get incident to different sums of labels. We are interested in the minimum k such that G admits such a k-labelling. We suspect that almost all graphs G can be labelled this way using labels $$1,\dots ,\Delta (G)$$ . Towards this speculation, we provide bounds on the maximum label value needed in general. In particular, we prove that using labels $$1,\dots ,\Delta (G)$$ is indeed sufficient when G is a tree, a particular cactus, or when its injective chromatic number $$\mathrm{\chi _{\mathrm{i}}}(G)$$ is equal to $$\Delta (G)$$ .