Magic and antimagic -decompositions

A decomposition of a graph G into isomorphic copies of a graph H is H -magic if there is a bijection 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 H in the decomposition is constant. It is known that complete graphs do not admit K 2 -magic decompositions for n > 6 . By using the results on the sumset partition problem, we show that the complete graph K 2 m + 1 admits T -magic decompositions by any graceful tree with m edges. We address analogous problems for complete bipartite graphs and for antimagic and ( a , d ) -antimagic decompositions.