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List vertex arboricity of planar graphs with \(5\)-cycles not adjacent to \(3\)-cycles and \(4\)-cycles. (English) Zbl 1424.05108
Summary: 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.
05C15 Coloring of graphs and hypergraphs
05C10 Planar graphs; geometric and topological aspects of graph theory
05C38 Paths and cycles