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arXiv · 1502.04755

Decomposition of Sparse Graphs into Forests: The Nine Dragon Tree Conjecture for $k \le 2$

Min Chen·Seog-Jin Kim·Alexandr Kostochka·Douglas B. West·Xuding Zhu
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Abstract

For a loopless multigraph $G$, the fractional arboricity $Arb(G)$ is the maximum of $\frac{|E(H)|}{|V(H)|-1}$ over all subgraphs $H$ with at least two vertices. Generalizing the Nash-Williams Arboricity Theorem, the Nine Dragon Tree Conjecture asserts that if $Arb(G)\le k+\frac{d}{k+d+1}$, then $G$ decomposes into $k+1$ forests with one having maximum degree at most $d$. The conjecture was previously proved for $d=k+1$ and for $k=1$ when $d \le 6$. We prove it for all $d$ when $k \le 2$, except for $(k,d)=(2,1)$.

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Min Chen, Seog-Jin Kim, Alexandr Kostochka, Douglas B. West, Xuding Zhu. 2015-02-17. Decomposition of Sparse Graphs into Forests: The Nine Dragon Tree Conjecture for $k \le 2$. https://arxiv.org/abs/1502.04755

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