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 $G$, $chi_l(G)leq left lceil frac{Delta(G)+1}{2}rightrceil$ where $Delta(G)$ is the maximum degree of $G$. First, 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 in some classes of graphs like cubic graphs and triangle-free planar graphs for which the conjecture was already known to be true. Next, for every 2-degenerate graph $G$, we show that $chi_l(G)=leftlceilfrac{Delta(G)}{2}rightrceil$ if $Delta(G)geq 5$. We conjecture that this equality holds also when $Delta(G)in{3,4}$ and show that this is the case for some well-known subclasses of 2-degenerate graphs. All our proofs can be converted into linear time algorithms.
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