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.
@article{bwmeta1.element.bwnjournal-article-doi-10_7151_dmgt_1434,
author = {Oleg V. Borodin and Anna O. Ivanova and Alexandr V. Kostochka and Naeem N. Sheikh},
title = {Decompositions of quadrangle-free planar graphs},
journal = {Discussiones Mathematicae Graph Theory},
volume = {29},
year = {2009},
pages = {87-99},
zbl = {1181.05034},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_7151_dmgt_1434}
}
Oleg V. Borodin; Anna O. Ivanova; Alexandr V. Kostochka; Naeem N. Sheikh. Decompositions of quadrangle-free planar graphs. Discussiones Mathematicae Graph Theory, Tome 29 (2009) pp. 87-99. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_7151_dmgt_1434/
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