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.

无向图 \(G = (V,E)\) 的树木性\( \Gamma (G) \)是使E可以划分为k 个森林的最小数k。纳什-威廉姆斯公式指出\(k = \lceil \gamma (G) \rceil \)，其中\(\gamma (G)\)是\(|E_H|/(|V_H| -1)的最大值\)覆盖G的所有子图\((V_H, E_H)\)和\(|V_H| \ge 2\)。强九龙树猜想指出，如果\(\gamma (G) \le k + \frac{d}{d+k+1}\)对于\(k, d \in {\mathbb {N}}_0 \)，则存在边集的划分G进入\(k+1\) 个森林，使得一个森林在每个连接的组件中最多有d 个边。我们对\(d \le k + 1\)进行猜想。对于\(d \le 2(k+1)\)，我们无法证明这个猜想，但是我们证明存在一个分区，其中一个森林中的连通分量最多有\(d + \lceil k \cdot \ frac{d}{k+1}\rceil - k\)边。作为该定理的应用，我们证明每个 5 边连通的平面图G都有一个\(\frac{5}{6}\)薄生成树。这个定理是最好的，因为我们不能用 4 边连接替换 5 边连接，即使我们替换\(\frac{5}{6}\)任何小于 1 的正实数。这强化了 Merker 和 Postle 的结果，该结果表明 6 边相连的平面图具有\(\frac{18}{19}\)薄生成树。