Geodesic
Linear arboricity of 1-planar graphs
Discussiones Mathematicae. Graph Theory, Tome 44 (2024) no. 2, pp. 435-457

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The linear arboricity la (G) of a graph G is the minimum number of linear forests that partition the edges of G. In 1981, Akiyama, Exoo and Harary conjectured that ⌈Δ(G)2⌉≤la (G) ≤⌈Δ(G)+12⌉ for any simple graph G. A graph G is 1-planar if it can be drawn in the plane so that each edge has at most one crossing. In this paper, we confirm the conjecture for 1-planar graphs G with Δ(G)≥13.
Keywords: linear arboricity, 1-planar graph, linear coloring, 3-alternating cycle 
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Wang, Weifan; Liu, Juan; Wang, Yiqiao. Linear arboricity of 1-planar graphs. Discussiones Mathematicae. Graph Theory, Tome 44 (2024) no. 2, pp. 435-457. http://geodesic.mathdoc.fr/item/DMGT_2024_44_2_a1/