On 3-choosability of planar graphs without certain cycles

Abstract

A graph G = ( V , E ) is L-colorable if for a given list assignment L = { L ( v ) : v ∈ V ( G ) } , 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.