Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Decomposition of Sparse Graphs into Forests: The Nine Dragon Tree
  Conjecture for $k \le 2$   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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The categories of this paper are: nlin
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The categories of this paper are: cmp-lg
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The categories of this paper are: astro-ph
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The categories of this paper are: q-fin
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The categories of this paper are: chao-dyn
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The categories of this paper are: plasm-ph
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The categories of this paper are: atom-ph
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The categories of this paper are: patt-sol
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The categories of this paper are: chem-ph
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The categories of this paper are: hep-ph
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The categories of this paper are: mtrl-th
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The categories of this paper are: nucl-ex
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The categories of this paper are: eess
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The categories of this paper are: math-ph
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The categories of this paper are: funct-an
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The categories of this paper are: alg-geom