The Overfull Nine Dragon Tree Conjecture is True
Mies, Sebastian1; Moore, Benjamin R.2
1 Johannes Gutenberg University Institute of Computer Science Mainz, Germany
2 Institute of Science and Technology Combinatorics and Probablity Group Klosterneuburg Austria
Innovations in Graph Theory, Volume 1 (2024), pp. 21-32.

Chen, Kim, Kostochka, West, and Zhu conjectured a strengthening of the Nine Dragon Tree Theorem that every graph that is (k,d)-sparse and has no overfull set decomposes into k+1 forests such that one of the forests has maximum degree d. Intuitively, the conjecture says that while the fractional arboricity bound in the Nine Dragon Tree Theorem is tight, we can relax the bound very slightly and still get the same result, so long as the obvious obstruction does not occur - that we can still decompose into k+1 forests. We prove this conjecture.

Received:
Accepted:
Published online:
DOI: 10.5802/igt.2
Classification: 05C70
Keywords: Nine Dragon Tree, Fractional arboricity, $(k,d)$-sparse
Mies, Sebastian 1; Moore, Benjamin R. 2

1 Johannes Gutenberg University Institute of Computer Science Mainz, Germany
2 Institute of Science and Technology Combinatorics and Probablity Group Klosterneuburg Austria
License: CC-BY 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights 
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Mies, Sebastian; Moore, Benjamin R. The Overfull Nine Dragon Tree Conjecture is True. Innovations in Graph Theory, Volume 1 (2024), pp. 21-32. doi : 10.5802/igt.2. https://igt.centre-mersenne.org/articles/10.5802/igt.2/

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