1-2 Conjecture for total sum-labellings

Prove that every graph admits a sum-proper total 2-labelling, meaning a labelling of all vertices and edges with labels from {1,2} such that adjacent vertices have distinct sums of the labels incident to them, including their own vertex labels.

Background

The paper studies total variants of the 1-2-3 Conjecture, in which both vertices and edges receive labels and each vertex is distinguished from its neighbours by an invariant derived from the labels incident to it. In the sum version, the invariant is the sum of the labels on the vertex and its incident edges.

The conjecture is known for several graph classes, including trees, complete graphs, 3-colourable graphs, and 4-regular graphs. The paper establishes additional results for certain bounded-degree classes, but does not resolve the conjecture for arbitrary graphs; the authors explicitly describe the sum and product versions as still widely open.