Decompositions of quadrangle-free planar graphs

Research supported in part by the RFBR grants 08-01-00673 and 06-01-00694. Research supported in part by the RFBR grants 08-01-00673 and 06-01-00694. Research supported in part by NSF grant DMS-0650784 and the RFBR grant 06-01-00694. abs W. He et al. showed that a planar graph not containing 4-cycles can be decomposed into a forest and a graph with maximum degree at most 7. This degree restriction was improved to 6 by Borodin et al. We further lower this bound to 5 and show that it cannot be improved to 3.