The Strong Nine Dragon Tree Conjecture is true for $d \leq k+1$

Abstract

The arboricity $\Gamma(G)$ of an undirected graph $G = (V,E)$ is the minimal number such that $E$ can be partitioned into $\Gamma(G)$ forests. Nash-Williams' formula states that $k = \lceil \gamma(G) \rceil$, where $\gamma(G)$ is the maximum of ${|E_H|}/(|V_H| -1)$ 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 $k, d \in \mathbb N_0$, then there is a partition of the edge set of $G$ into $k+1$ forests such that one forest has at most $d$ edges in each connected component. We settle the conjecture for $d \leq k + 1$. For $d \leq 2(k+1)$, we cannot prove the conjecture, however we show that there exists a partition in which the connected components in one forest have at most $d + \lceil k \cdot \frac{d}{k+1} \rceil - k$ edges. As an application of this theorem, we show that every $5$-edge-connected planar graph $G$ has a $\frac{5}{6}$-thin spanning tree. This theorem is best possible, in the sense that we cannot replace $5$-edge-connected with $4$-edge-connected, even if we replace $\frac{5}{6}$ with any positive real number less than $1$. This strengthens a result of Merker and Postle which showed $6$-edge-connected planar graphs have a $\frac{18}{19}$-thin spanning tree.