The Nine Dragon Tree Conjecture for bounded forest decompositions

Let GG be a graph, and let k,dk,d be nonnegative integers. The Nine Dragon Tree Conjecture. If

γf(G)k+dd+k+1,\gamma_f(G)\leq k+\frac{d}{d+k+1},

then GG decomposes into k+1k+1 forests, one of which is dd-bounded. The conjecture is the graph-theoretic Nine Dragon Tree statement concerning decomposition into forests with one bounded component. It was proved by Jiang and Yang, so the conjecture is resolved.

Sources & referencesView supporting material

Primary source

Hui Gao, “Packing spanning arborescences with extra large one”, arXiv:2511.18952 (2025).

Additional references

3 papers in this index state this conjecture (2016–2025). The statement above is taken from the most recent of them; the others are arXiv:2201.10791, arXiv:1608.05352.

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