Magic and antimagic HH-decompositions

A decomposition of a graph GG into isomorphic copies of a graph HH is HH-magic if there is a bijection f:V(G)∪E(G)→{0,1,…,|V(G)|+|E(G)|−1}f:V(G)∪E(G)→{0,1,…,|V(G)|+|E(G)|−1} such that the sum of labels of edges and vertices of each copy of HH in the decomposition is constant. It is known that complete graphs do not admit K2K2-magic decompositions for n>6n>6. By using the results on the sumset partition problem, we show that the complete graph K2m+1K2m+1 admits TT-magic decompositions by any graceful tree with mm edges. We address analogous problems for complete bipartite graphs and for antimagic and (a,d)(a,d)-antimagic decompositions.