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

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