1-2 Conjecture for total product-labellings

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

Background

The product version replaces the incident-label sum by the product of the labels on a vertex and its incident edges. The paper notes that deciding whether a graph admits a product-proper total 2-labelling is NP-complete, while the conjectured bound is known for several classes, including complete graphs, 3-colourable graphs, and 4-regular graphs.

The paper proves the product conjecture for graphs of maximum degree at most 6, but the general conjecture remains unresolved. The authors explicitly state that the sum and product versions are still widely open.