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 \(\Delta (G)\).

在这项工作中，我们引入并研究了一个新的图形标注问题，该问题是1-2-3猜想和图形的内射着色的结合，也发现了与图形不规则性概念的联系。在这个问题中，给定图形G的目标是标记G的边缘，以使具有共同邻居的每两个顶点入射到不同的标记和上。我们对最小k感兴趣，以 使G接受这样的k标签。我们怀疑几乎所有图形G都可以使用标签\（1，\ dots，\ Delta（G）\）进行标注。为了进行这种推测，我们通常会限制所需的最大标签值。特别是，我们证明，当G是一棵树，一个特定的仙人掌或当其内射色数\（\ mathrm {\ chi _ {时，使用标签\（1，\ dots，\ Delta（G）\）确实足够\ mathrm {i}}}（G）\）等于\（\ Delta（G）\）。