# Two short proofs of the Perfect Forest Theorem

• Published in 2016
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A perfect forest is a spanning forest of a connected graph $G$, all of whose components are induced subgraphs of $G$ and such that all vertices have odd degree in the forest. A perfect forest generalised a perfect matching since, in a matching, all components are trees on one edge. Scott first proved the Perfect Forest Theorem, namely, that every connected graph of even order has a perfect forest. Gutin then gave another proof using linear algebra. We give here two very short proofs of the Perfect Forest Theorem which use only elementary notions from graph theory. Both our proofs yield polynomial-time algorithms for finding a perfect forest in a connected graph of even order.

### BibTeX entry

@article{TwoshortproofsofthePerfectForestTheorem,
title = {Two short proofs of the Perfect Forest Theorem},
abstract = {A perfect forest is a spanning forest of a connected graph {\$}G{\$}, all of whose
components are induced subgraphs of {\$}G{\$} and such that all vertices have odd
degree in the forest. A perfect forest generalised a perfect matching since, in
a matching, all components are trees on one edge. Scott first proved the
Perfect Forest Theorem, namely, that every connected graph of even order has a
perfect forest. Gutin then gave another proof using linear algebra.
We give here two very short proofs of the Perfect Forest Theorem which use
only elementary notions from graph theory. Both our proofs yield
polynomial-time algorithms for finding a perfect forest in a connected graph of
even order.},
url = {http://arxiv.org/abs/1612.05004v1 http://arxiv.org/pdf/1612.05004v1},
author = {Yair Caro and Josef Lauri and Christina Zarb},
comment = {},
urldate = {2016-12-18},
archivePrefix = {arXiv},
eprint = {1612.05004},
primaryClass = {math.CO},
year = 2016,
collections = {About proof,Fun maths facts}
}