The linear arboricity of planar graphs with no short cycles

AbstractThe linear arboricity of a graph G is the minimum number of linear forests which partition the edges of G. Akiyama, Exoo and Harary conjectured that ⌈Δ(G)2⌉≤la(G)≤⌈Δ(G)+12⌉ for any simple graph G. In the paper, it is proved that if G is a planar graph with Δ≥7 and without i-cycles for some i∈{4,5}, then la(G)=⌈Δ(G)2⌉

The Linear Arboricity of Planar Graphs with Maximum Degree at Least Five

Let G be a planar graph with maximum degree ∆ ≥ 5. It is proved that la(G) = ∆(G)/2 if (1) any 4-cycle is not adjacent to an i-cycle for any i ∈ {3, 4, 5} or (2) G has no intersecting 4-cycles and intersecting i-cycles for some i ∈ {3, 6}