1-2 Conjecture for total multiset-labellings

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

Background

The multiset version distinguishes adjacent vertices by the multiset of labels incident to each vertex, including the label assigned to the vertex itself. Because vertices of different degrees automatically yield multisets of different cardinalities, the authors regard multiset distinction as easier than sum or product distinction.

The multiset conjecture is introduced in this paper and is known for graph classes covered by corresponding sum or product results, including complete graphs, 3-colourable graphs, and 4-regular graphs. The paper proves it for graphs of maximum average degree at most 3 and, as a consequence of its product results, for graphs of maximum degree at most 6, but leaves the general conjecture unresolved.