List vertex arboricity of planar graphs with 5-cycles not adjacent to 3-cycles and 4-cycles

A graph G is list k-arborable if for any sets L(v) of cardinality at least k at its vertices, one can choose an element (color) for each vertex v from its list L(v) so that the subgraph induced by every color class is an acyclic graph (a forest). In the paper, it is proved that every planar graph with 5-cycles not adjacent to 3-cycles and 4-cycles is list 2-arborable.