Further results on an equitable 1-2-3 Conjecture

In this work, we consider equitable proper labellings of graphs, which were recently introduced by Baudon, Pilśniak, Przybyło, Senhaji, Sopena, and Woźniak. Given a graph G, the goal is to assign labels edges so that (1) no two adjacent vertices are incident same sum labels, (2) every assigned about number times. Particularly, aim at designing such k-labellings G with k being as small possible. connection so-called 1-2-3 Conjecture, it might be 1,2,3 are, few obvious exceptions apart, always sufficient achieve just in non-equitable version problem. We provide results regarding some open questions labellings. Via hardness result, first prove there exist infinitely many graphs for more required than version. This remains true bipartite case. finally show that, k≥3, k-regular admits an k-labelling.