Further evidence towards the multiplicative 1-2-3 Conjecture

The product version of the 1-2-3 Conjecture, introduced by Skowronek-Kaziów in 2012, states that, a few obvious exceptions apart, all graphs can be 3-edge-labelled so that no two adjacent vertices get incident to same labels. To date, this conjecture was mainly verified for complete and 3-colourable graphs. As strong support conjecture, it also proved admit such 4-labellings. In work, we investigate how recent proof multiset Conjecture Vu?kovi? adapted prove results on version. We 4-chromatic verify Conjecture. design 3-labellings almost have desired property. This leads new problem, solve some graph classes.