Geodesic
Decompositions of quadrangle-free planar graphs
Discussiones Mathematicae. Graph Theory, Tome 29 (2009) no. 1, pp. 87-99

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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.
Keywords: planar graphs, graph decompositions, quadrangle-free graphs 
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Borodin, Oleg; Ivanova, Anna; Kostochka, Alexandr; Sheikh, Naeem. Decompositions of quadrangle-free planar graphs. Discussiones Mathematicae. Graph Theory, Tome 29 (2009) no. 1, pp. 87-99. http://geodesic.mathdoc.fr/item/DMGT_2009_29_1_a5/