On the total choosability of planar graphs and of sparse graphs

In the current paper, we prove the 11-total choosability of planar graphs with maximum degree Δ⩽8, the (Δ+1)-total choosability of 5-cycle-free planar graphs with maximum degree Δ⩾8, the 5-total choosability of graphs with maximum degree Δ=4 and maximum average degree mad<3, and the 4-total choosability of subcubic graphs with maximum average degree .