Gyárfás's tree packing conjecture

Take any trees $T_2,T_3,\dots,T_n$ where $T_k$ has $k$ vertices. Their edge counts add up to $1+2+\dots+(n-1)=\binom n2$, exactly the number of edges of the complete graph $K_n$. Gyárfás conjectured in 1976 that the trees can always be fitted together to decompose $K_n$, whatever their shapes. The conjecture is now known for trees of bounded degree and for the largest trees, but not in general.

Problem

Let $T_2,\dots,T_n$ be trees such that $T_k$ has exactly $k$ vertices. Can the edge set of the complete graph $K_n$ always be written as a disjoint union of copies of $T_2,\dots,T_n$, one copy of each?

Background

The conjecture is due to Gyárfás (1976) and appears in Erdős's 1981 list of combinatorial problems he most wanted to see solved; it is often stated with Lehel as co-author. Since the edge counts match exactly, the question is whether the trees can always be embedded without overlapping, a "perfect packing". Gyárfás and Lehel (1978) proved it when all but at most two of the trees are stars, or when every tree is a star or a path.

What is known

• Small cases. Fishburn (1983) verified the conjecture for $n\le9$, and Guichard and Massman (1990) extended a computer verification through $n=11$. An exhaustive check for $n=12$, with the method and its limits described, was posted on the problem's forum in September 2026; it is not yet published.
• Greedy packing. Bollobás (1983) showed that the smallest $\lfloor n/\sqrt2\rfloor$ trees can always be packed greedily into $K_n$.
• Bounded degree. Joos, Kim, Kühn and Osthus (2019) proved the conjecture for trees of bounded maximum degree, and Allen, Böttcher, Clemens, Hladký, Piguet and Taraz extended it to maximum degree at most $cn/\log n$.
• Largest trees. Janzer and Montgomery (2024) proved that the largest $cn$ trees of any such sequence can always be packed into $K_n$, confirming a conjecture of Bollobás.
• Claims. A full proof claimed by Chalise, Clark and Gnang (arXiv:2410.13840, 2024) has not been accepted; people who tried to verify it, and checks reported on the problem's forum, found serious gaps.

What would count as progress

• A counterexample: a sequence of trees that cannot be packed into $K_n$, which can be confirmed by an exhaustive search for that sequence.
• An independent, published exhaustive check for $n=12$, or a first one for $n=13$, with code that others can rerun.
• A proof for new classes of trees, for example with maximum degree up to $cn$.
• A formal proof against the Lean statement in formal-conjectures.

References

1. P. C. Fishburn, Balanced integer arrays: a matrix packing theorem, J. Combin. Theory Ser. A 34 (1983), 98–101. doi:10.1016/0097-3165(83)90045-6
2. B. Bollobás, Some remarks on packing trees, Discrete Math. 46 (1983), 203–204. doi:10.1016/0012-365X(83)90254-6
3. F. Joos, J. Kim, D. Kühn and D. Osthus, Optimal packings of bounded degree trees, J. Eur. Math. Soc. 21 (2019), 3573–3647. doi:10.4171/JEMS/909
4. P. Allen, J. Böttcher, D. Clemens, J. Hladký, D. Piguet and A. Taraz, The tree packing conjecture for trees of almost linear maximum degree (2021). arXiv:2106.11720
5. B. Janzer and R. Montgomery, Packing the largest trees in the tree packing conjecture (2024). arXiv:2403.10515
6. D. R. Guichard and J. D. Massman, A note on packing complete graphs with trees, J. Combin. Math. Combin. Comput. 8 (1990), 123–126. combinatorialpress.com

Source

Imported from Erdős Problems 743, curated by Thomas Bloom. Open according to the source as of 2026-10-20. The statement above is written in our own words; see the source for the original wording, history and discussion.