A note on divisibility of the number of matchings of a family of graphs.
Decompositions of quadrangle-free planar graphs
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.
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