Title: 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.