(Δ+1Δ+1)-total choosability of planar graphs with no cycles of length from 4 to kk and without close triangles

Let GG be a planar graph with maximum degree Δ(G)Δ(G). In this paper, we prove that GGis (Δ(G)+1Δ(G)+1)-total choosable if GG has no cycle of length from 44 to kk and has minimum distance at least dΔdΔ between triangles for (Δ(G),k,dΔ)=(6,4,1),(5,5,2),(5,6,1),(5,7,0),(4,6,3)(Δ(G),k,dΔ)=(6,4,1),(5,5,2),(5,6,1),(5,7,0),(4,6,3), (4,7,2),(4,10,1)(4,7,2),(4,10,1).

► We study (D+1)(D+1)-total choosability of planar graphs with maximum degree DD. ► The absence of cycle of length from 4 to kk and of triangles at distance less than dd is sufficient. ► The cases (D,k,d)=(6,4,1),(5,5,2),(5,6,1),(5,7,0),(4,6,3),(4,7,2),(4,10,1)(D,k,d)=(6,4,1),(5,5,2),(5,6,1),(5,7,0),(4,6,3),(4,7,2),(4,10,1) are proved. ► These results improve and generalize recent results from Borodin et al., Chang et al., and Hou et al. ► The proofs rely on the structural properties of an hypothetical minimal counter example.