The Strong Nine Dragon Tree Conjecture is True for $d \leq 2(k+1)$
Abstract
The arboricity $\Gamma(G)$ of an undirected graph $G =(V,E)$ is the minimal number $k$ such that $E$ can be partitioned into $k$ forests on $V$. NashWilliams' formula states that $k = \lceil \gamma(G) \rceil$, where $\gamma(G)$ is the maximum of $\frac{E_{H}}{V_{H}1}$ over all subgraphs $(V_H , E_H )$ of $G$ with $V_H  \geq 2$. The Strong Nine Dragon Tree Conjecture states that if $\gamma(G) \leq k + \frac{d}{d+k+1}$ for $k, d \in \mathbb{N}$, then there is a partition of the edge set of $G$ into $k + 1$ forests on $V$ such that one forest has at most $d$ edges in each connected component. Here we prove the Strong Nine Dragon Tree Conjecture when $d \leq 2(k +1)$, which is a new result for all $(k, d)$ such that $d > k + 1$. In fact, we prove a stronger theorem. We prove that a weaker sparsity notion, called $(k, d)$sparseness, suffices to give the decomposition, under the assumption that the graph decomposes into $k+1$ forests. This is a new result for all $(k, d)$ where $d > 1$, and improves upon the recent resolution of the Overfull Nine Dragon Tree Theorem for all $(k, d)$ when $d \leq 2(k +1)$. As a corollary, we obtain that planar graphs of girth five decompose into a forest and a forest where every component has at most four edges, and by duality, we obtain that $5$edgeconnected planar graphs have a $\frac{4}{5}$thin tree, improving a result of the authors that $5$edgeconnected planar graphs have a $\frac{5}{6}$thin tree
 Publication:

arXiv eprints
 Pub Date:
 March 2024
 DOI:
 10.48550/arXiv.2403.05178
 arXiv:
 arXiv:2403.05178
 Bibcode:
 2024arXiv240305178M
 Keywords:

 Mathematics  Combinatorics
 EPrint:
 33 pages, multiple figures