General 1-2 Conjecture for total multiset-labelings

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

Background

The multiset version distinguishes adjacent vertices by the multiset of labels on their incident elements, including each vertex's own label. Because vertices of different degrees automatically have multisets of different cardinalities, multiset distinction is generally easier than sum or product distinction.

The authors introduce this variant formally because, to their knowledge, it had not previously been stated as a conjecture. Known results for the sum and product versions imply it for several classes, and the best general bound cited in the introduction is xΜ(G) ≤ 3. The paper proves the conjecture for graphs of maximum average degree at most 3 and obtains further bounded-degree consequences, but the unrestricted conjecture remains open.