On Inducing Degenerate Sums Through 2-Labellings

We deal with a variant of the 1–2–3 Conjecture introduced by Gao, Wang, and Wu (Graphs Combin 32:1415–1421, 2016) . This variant asks whether all graphs can have their edges labelled with 1 and 2 so that when computing the sums of labels incident to the vertices, no monochromatic cycle appears. In the aforementioned seminal work, the authors mainly verified their conjecture for a few classes of graphs, namely graphs with maximum average degree at most 3 and series–parallel graphs, and observed that it also holds for simple classes of graphs (cycles, complete graphs, and complete bipartite graphs). In this work, we provide a deeper study of this conjecture, establishing strong connections with other, more or less distant notions of graph theory. While this conjecture connects quite naturally to other notions and problems surrounding the 1–2–3 Conjecture, it can also be expressed so that it relates to notions such as the vertex-arboricity of graphs. Exploiting such connections, we provide easy proofs that the conjecture holds for bipartite graphs and 2-degenerate graphs, thus generalising some of the results of Gao, Wang, and Wu. We also prove that the conjecture holds for graphs with maximum average degree less than \(\frac{10}{3}\), thereby strengthening another of their results. Notably, this also implies the conjecture holds for planar graphs with girth at least 5. All along the way, we also raise observations and results highlighting why the conjecture might be of greater interest.