Equitable neighbour-sum-distinguishing edge and total colourings

With any (not necessarily proper) edge k-colouring :E(G){1,,k} of a graph G, one can associate a vertex colouring given by (v)=ev(e). A neighbour-sum-distinguishing 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 extend 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 for which the number of elements in any two colour classes of differ by at most one. We determine the equitable neighbour-sum-distinguishing index of complete graphs, complete bipartite graphs and forests, and the equitable neighbour-sum-distinguishing total chromatic number of complete graphs and bipartite graphs.