Equitable neighbour-sum-distinguishing edge and total colourings

Abstract

With any (not necessarily proper) edge $k$-colouring $\gamma:E(G)\longrightarrow\{1,\dots,k\}$ of a graph $G$,one can associate a vertex colouring $\sigma\_{\gamma}$ given by $\sigma\_{\gamma}(v)=\sum\_{e\ni v}\gamma(e)$.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 $G$ admitsa 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 andtotal colourings, that is colourings $\gamma$ for whichthe number of elements in any two colour classes of $\gamma$ differ by at most one.We determine the equitable neighbour-sum-distinguishing indexof complete graphs, complete bipartite graphs and forests,and the equitable neighbour-sum-distinguishing total chromatic numberof complete graphs and bipartite graphs.