The linear arboricity conjecture for 3-degenerate graphs.

A $k$-linear coloring of a graph $G$ is an edge coloring of $G$ with $k$ colors so that each color class forms a linear forest---a forest whose each connected component is a path. The linear arboricity $\chi_l'(G)$ of $G$ is the minimum integer $k$ such that there exists a $k$-linear coloring of $G$. Akiyama, Exoo and Harary conjectured in 1980 that for every graph $G$, $\chi_l'(G)\leq \left \lceil \frac{\Delta(G)+1}{2}\right\rceil$ where $\Delta(G)$ is the maximum degree of $G$. We prove the conjecture for 3-degenerate graphs. This establishes the conjecture for graphs of treewidth at most 3 and provides an alternative proof for the conjecture for triangle-free planar graphs. Our proof also yields an $O(n)$-time algorithm that partitions the edge set of any 3-degenerate graph $G$ on $n$ vertices into at most $\left\lceil\frac{\Delta(G)+1}{2}\right\rceil$ linear forests. Since $\chi'_l(G)\geq\left\lceil\frac{\Delta(G)}{2}\right\rceil$ for any graph $G$, the partition produced by the algorithm differs in size from the optimum by at most an additive factor of 1.