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

TL;DR

This paper proves the Nine Dragon Tree Conjecture for sparse graphs with certain parameters, showing they can be decomposed into forests with degree constraints, extending previous partial results.

Contribution

It establishes the conjecture for all cases with k ≤ 2, except one, advancing understanding of graph decompositions into forests.

Findings

Proved the conjecture for all d when k ≤ 2, except (2,1)

Extended the class of graphs known to satisfy the conjecture

Provided new decomposition methods for sparse multigraphs

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 $(k,d)=(2,1)$.