Generalizations of the Tree Packing Conjecture

The Gy'arf'as tree packing conjecture asserts that any set of trees with $2,3,\mathrm{.}\mathrm{.}\mathrm{.},k$ vertices has an (edge-disjoint) packing into the complete graph on $k$ vertices. Gy'arf'as and Lehel proved that the conjecture holds in some special cases. We address the problem of packing trees into $k$-chromatic graphs. In particular, we prove that if all but three of the trees are stars then they have a packing into any $k$-chromatic graph. We also consider several other generalizations of the conjecture.