Equitable neighbour-sum-distinguishing edge and total colourings

With any (not necessarily proper) edge k-colouring γ : E(G) −→ {1, . . . , k} of a graph G, one can associate a vertex colouring σγ given by σγ(v) = ∑ e∋v γ(e). A neighbour-sumdistinguishing edge k-colouring is an edge colouring whose associated vertex colouring is proper. The neighbour-sum-distinguishing index of a graph G is then the smallest k for which G admits a neighbour-sum-distinguishing edge k-colouring. These notions naturally extends to total colourings of graphs that assign colours to both vertices and edges. We study in this paper equitable neighbour-sum-distinguishing edge colourings and total colourings, that is colourings γ for which the number of elements in any two colour classes of γ differ by at most one. We determine the equitable neighbour-sumdistinguishing index of complete graphs, complete bipartite graphs and forests, and the equitable neighbour-sum-distinguishing total chromatic number of complete graphs and bipartite graphs.