General 1-2 Conjecture for total sum-labelings

Prove that every graph admits a sum-proper total 2-labelling, equivalently that the total sum-distinguishing parameter satisfies xΣ(G) ≤ 2 for every graph G.

Background

The total 1-2 Conjecture is the total-labeling counterpart of the 1-2-3 Conjecture: vertices and edges are assigned labels from {1,2}, and each vertex is distinguished from its neighbors by the sum of the labels on its incident elements, including its own vertex label. The conjecture is motivated by the observation that total labelings may require fewer labels than edge-only labelings.

The paper notes that the conjecture is known for several graph classes, including trees, complete graphs, 3-colourable graphs, and 4-regular graphs, while deciding whether a graph admits a sum-proper total 2-labelling is NP-complete in general. The general conjecture remains unresolved, and the paper contributes results for additional restricted classes through related product and multiset variants.