The linear arboricity conjecture for 3-degenerate graphs
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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 χ_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, χ_l'(G)≤⌈Δ(G)+1/2⌉ where Δ(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 ⌈Δ(G)+1/2⌉ linear forests. Since χ'_l(G)≥⌈Δ(G)/2⌉ for any graph G, the partition produced by the algorithm differs in size from the optimum by at most an additive factor of 1.
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