AVD-total-colouring of complete equipartite graphs

An AVD-total-colouring   of a simple graph GG is a mapping π:V(G)∪E(G)→Cπ:V(G)∪E(G)→C, CC a set of colours, such that: (i) for each pair of adjacent or incident elements x,y∈V(G)∪E(G)x,y∈V(G)∪E(G), π(x)≠π(y)π(x)≠π(y); (ii) for each pair of adjacent vertices x,y∈V(G)x,y∈V(G), sets {π(x)}∪{π(xv):xv∈E(G),v∈V(G)}{π(x)}∪{π(xv):xv∈E(G),v∈V(G)} and {π(y)}∪{π(yv):yv∈E(G),v∈V(G)}{π(y)}∪{π(yv):yv∈E(G),v∈V(G)} are distinct. The AVD-total-chromatic number  , χa″(G), is the smallest number of colours for which GG admits an AVD-total-colouring. In 2005, Zhang et al. conjectured that χa″(G)≤Δ(G)+3 for any simple graph GG. In this article this conjecture is verified for any complete equipartite graph. Moreover, if GG is a complete equipartite graph of even order, then it is shown that χa″(G)=Δ(G)+2.