The Strong Nine Dragon Tree Conjecture is True for $d \le k + 1$

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. 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 $|V_H| \ge 2$. The Strong Nine Dragon Tree Conjecture states that if $\gamma (G) \le k + \frac{d}{d+k+1}$ 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 $d \le k + 1$. For $d \le 2(k+1)$, 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.