In 1980, Akiyama, Exoo and Harary posited the Linear Arboricity Conjecture which states that any graph G of maximum degree \(\Delta \) can be decomposed into at most linear forests. (A forest is linear if all of its components are paths.) In 1988, Alon proved the conjecture holds asymptotically. The current best bound is due to Ferber, Fox and Jain from 2020 who showed that \(\frac{\Delta }{2}+ O(\Delta ^{0.661})\) suffices for large enough \(\Delta \). Here, we show that G admits a decomposition into at most \(\frac{\Delta }{2}+ 3\sqrt{\Delta } \log ^4 \Delta \) linear forests provided \(\Delta \) is large enough. Moreover, our result also holds in the more general list setting, where edges have (possibly different) sets of permissible linear forests. Thus our bound also holds for the List Linear Arboricity Conjecture which was only recently shown to hold asymptotically by Kim and the second author. Indeed, our proof method ties together the Linear Arboricity Conjecture and the well-known List Colouring Conjecture; consequently, our error term for the Linear Arboricity Conjecture matches the best known error-term for the List Colouring Conjecture due to Molloy and Reed from 2000. This follows as we make two copies of every colour and then seek a proper edge colouring where we avoid bicoloured cycles between a colour and its copy; we achieve this via a clever modification of the nibble method.

1980 年，Akiyama、Exoo 和 Harary 提出了线性 Arboricity 猜想，该猜想指出任何最大度\(\Delta \)的图G最多可以分解为线性森林。（如果森林的所有组成部分都是路径，则森林是线性的。）1988 年，Alon 证明了该猜想渐近成立。当前的最佳界限归功于 2020 年的 Ferber、Fox 和 Jain，他们表明\(\frac{\Delta }{2}+ O(\Delta ^{0.661})\)足以满足足够大的\(\Delta \) . 在这里，我们表明G允许分解为最多\(\frac{\Delta }{2}+ 3\sqrt{\Delta } \log ^4 \Delta \) 个线性森林，提供\(\Delta \)足够大。此外，我们的结果也适用于更一般的列表设置，其中边具有（可能不同的）允许线性森林集。因此，我们的界限也适用于 List Linear Arboricity 猜想，该猜想最近才被 Kim 和第二作者证明渐近成立。事实上，我们的证明方法将线性树枝猜想和著名的列表着色猜想联系在一起；因此，我们的线性 Arboricity 猜想的错误项与 2000 年由 Molloy 和 Reed 提出的列表着色猜想的最著名错误项相匹配。这是因为我们对每种颜色制作两份副本，然后在我们避免的地方寻找适当的边缘着色一种颜色和它的副本之间的双色循环；我们通过巧妙地修改半字节方法来实现这一点。