The linear arboricity of a graph G, denoted by $\text{la}(G)$, is the minimum number of edge-disjoint linear forests (i.e. forests in which every connected component is a path) in G whose union covers all the edges of G. A famous conjecture due to Akiyama, Exoo, and Harary from 1981 asserts that $\text{la}(G)\le \lceil (\mathrm{\Delta }(G)+1)/2\rceil$, where $\mathrm{\Delta }(G)$ denotes the maximum degree of G. This conjectured upper bound would be best possible, as is easily seen by taking G to be a regular graph. In this paper, we show that for every graph G, $\text{la}(G)\le \frac{\mathrm{\Delta }}{2}+O({\mathrm{\Delta }}^{2/3-\alpha })$ for some $\alpha >0$, thereby improving the previously best known bound due to Alon and Spencer from 1992. For graphs which are sufficiently good spectral expanders, we give even better bounds. Our proofs of these results further give probabilistic polynomial time algorithms for finding such decompositions into linear forests.

图G的线性树率，表示为$\text{啦啦}（G）$是边缘不相交线性森林的最小数（即森林，其中每个连接的组件的路径）在ģ其联合盖的所有边缘ģ。著名的猜想是1981年的Akiyama，Exoo和Harary提出的$\text{啦啦}（G）\le \lceil （\mathrm{\Delta }（G）+\mathrm{1个}）/2\rceil$，在哪里 $\mathrm{\Delta }（G）$表示G的最大程度。通过将G设为正则图可以很容易地看出这个推测的上限是最好的。在本文中，我们表明对于每个图G，$\text{啦啦}（G）\le \frac{\mathrm{\Delta }}{2}+Ø（{\mathrm{\Delta }}^{2/3-\alpha }）$ 对于一些 $\alpha >0$，从而改善了1992年以来由Alon和Spencer引起的最著名的界线。对于足够好的谱扩展器的图，我们给出甚至更好的界线。我们对这些结果的证明进一步提供了概率多项式时间算法，用于发现这种分解成线性森林。