On 3-choosability of planar graphs without certain cycles

A graph G=(V,E) is L-colorable if for a given list assignment , there exists a proper coloring c of G such that c(v)∈L(v) for all v∈V. If G is L-colorable for every list assignment L with |L(v)|⩾k for all v∈V, then G is said to be k-choosable. In this paper, we prove that every planar graph with neither 5-, 6-, and 7-cycles nor triangles of distance less than 3, or with neither 5-, 6-, and 8-cycles nor triangles of distance less than 2 is 3-choosable.