# Tree packing conjectures; Graceful tree labeling conjecture

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A family of graphs $H_1,...,H_k$ packs into a graph $G$ if there exist pairwise edge-disjoint copies of $H_1,...,H_k$ in $G$. Gyarfas and Lehel conjectured that any family $T_1, ..., T_n$ of trees of respective orders $1, ..., n$ packs into $K_n$. A similar conjecture of Ringel asserts that $2n$ copies of any trees $T$ on $n+1$ vertices pack into $K_{2n+1}$.In a joint work with Boettcher, Piguet, Taraz we proved a theorem about packing trees. The theorem implies asymptotic versions of the above conjectures for families of trees of bounded maximum degree. Tree-indexed random walks controlled by the nibbling method are used in the proof. In a joint work with Adamaszek, Adamaszek, Allen and Grosu, we used the nibbling method to prove the approximate version of the related Graceful Tree Labeling conjecture for trees of bounded degree.