Planar Graphs Without 4-Cycles Adjacent to 3-Cycles Are List Vertex 2-Arborable

It is known that not all planar graphs are 4-choosable; neither all of them are vertex 2-arborable. However, planar graphs without 4-cycles and even those without 4-cycles adjacent to 3-cycles are known to be 4-choosable. We extend this last result in terms of covering the vertices of a graph by induced subgraphs of variable degeneracy. In particular, we prove that every planar graph without 4-cycles adjacent to 3-cycles can be covered by two induced forests. (C) 2009 Wiley Periodicals, Inc. J Graph Theory 62: 234-240, 2009