The linear arboricity of planar graphs with maximum degree at least 5

Let G   be a planar graph with maximum degree Δ(G)⩾5Δ(G)⩾5. It is proved that la(G)=⌈Δ(G)2⌉ if G has no intersecting 4-cycles and intersecting 5-cycles.

► Generalizing Tanʼs results (Tan et al., 2011 [10]) and prove that if G   is a planar graph with Δ(G)⩾5Δ(G)⩾5 and without intersecting 4-cycles and intersecting 5-cycles, then la(G)=⌈Δ(G)2⌉.
► Combining Eulerʼs formula with the linear arboricity.
► Constructing some new structures of the planar graphs.