On resolvable tree‐decompositions of complete graphs

A partition of the edge set of a graph H into subsets inducing graphs H 1 ,…, H s isomorphic to a graph G is said to be a G ‐decomposition of H . A G ‐decomposition of H is resolvable if the set { H 1 ,…, H s } can be partitioned into subsets, called resolution classes , such that each vertex of H occurs precisely once in each resolution class. We prove that for every graceful tree T of odd order the obvious necessary conditions for the existence of a resolvable T ‐decomposition of a complete graph are asymptotically sufficient. This generalizes the results of Horton and Huang concerning paths and stars.