A notion of vertex equitability for proper labellings

We introduce an equitable version of proper labellings of graphs, where the notion of equitability is with respect to the resulting vertex sums. That is, we are interested in k-labellings where, when computing the sums of labels incident to the vertices, we get a vertex-colouring that is not proper only, but also equitable. For a given graph G, we are interested in the parameter chi(Sigma)(G), which is the smallest k >= 1 (if any) such that G admits such k-labellings. Through examples of particular graph classes, we observe that this new parameter chi(Sigma) behaves sort of similarly to the parameters chi(Sigma) and s, whose parameters lie behind the 1-2-3 Conjecture and the irregularity strength of graphs, in a more or less strong way, depending on the graphs considered. We then prove general bounds on chi(Sigma), showing that, in some contexts (trees and connected graphs with large minimum degree), this parameter is bounded above by roughly 3n/4 for an n-graph. We also prove that determining chi(Sigma) is NP-hard in general, and finish off with directions for further work on the topic. (c) 2023 Elsevier B.V. All rights reserved.