Complexity of the H-factor problem.
An -factor in a graph is a set of vertex-disjoint copies of covering all vertices of .
The answer is positive for cliques and a few other graphs [KO06].
If we remove the minimum degree condition, the problem is NP-complete if and only if has a component which contains at least 3 vertices, as shown by Hell and Kirkpatrick [HK].
Bibliography
[HK] P. Hell and D.G. Kirkpatrick, On the complexity of general graph factor problems, SIAM J. Computing 12 (1983), 601-609.
[KO06] D. Kühn and D. Osthus, Critical chromatic number and the complexity of perfect packings in graphs, Proceedings of the 17th ACM-SIAM Symposium on Discrete Algorithms (SODA), 2006.
*[KO09] D. Kühn and D. Osthus, The minimum degree threshold for perfect graph packings, Combinatorica 29 (2009), 65-107.
* indicates original appearance(s) of problem.