On the Linear Arboricity of Graphs with Treewidth at Most Four

The linear arboricity la ( G ) of a graph G is the minimum number of linear forests that partition the edges of G . Akiyama, Exoo and Harary conjectured that ⌈ Δ 2 ⌉ ≤ l a ( G ) ≤ ⌈ Δ + 1 2 ⌉ for any graph G with maximum degree Δ , and proved the conjecture holds for forests. This conjecture has been verified for certain graph families with treewidth at most 3. In the paper we improve these former results by validating the conjecture for all graphs with treewidth at most 4.