List Vertex Arboricity of Planar Graphs with $5$-Cycles Not Adjacent to $3$-Cycles and $4$-Cycles

Abstract

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.