General 1-2 Conjecture for total product-labelings

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

Background

The product version replaces the incident-label sum at each vertex with the product of the labels on all incident elements, including the vertex label. A total 2-labelling is product-proper when adjacent vertices receive distinct incident-label products.

Before this paper, the product conjecture was known for several classes, such as complete and 3-colourable graphs, and for 4-regular graphs through equivalence with the sum formulation in regular graphs. The general problem is computationally difficult—testing for a product-proper total 2-labelling is NP-complete—and the paper establishes the conjecture for graphs of maximum degree at most 6 but does not resolve it for arbitrary graphs.