On the role of 3s for the 1-2-3 Conjecture

The 1-2-3 Conjecture states that every connected graph different from K2 admits a proper 3-(edge-)labelling, i.e., can have its edges labelled with 1,2,3 so no two adjacent vertices are incident to the same sum of labels. In connection some recent optimisation variants this conjecture, in paper we investigate role label 3 3-labellings graphs. An intuition previous investigations is that, general, it should always be possible produce assigning only few edges. We prove for p≥0, there various graphs needing at least p 3s their 3-labellings. Actually, deciding whether given properly 3-labelled NP-complete p≥0. also focus on classes 3-chromatic For such (cacti, cubic graphs, triangle-free planar etc.), p≥1 all admit most cases, provide lower and upper bounds number needed.