Two short proofs of the Perfect Forest Theorem

Authors

Yair Caro, University of Haifa-OranimFollow
Josef Lauri, University of MaltaFollow
Christina Zarb, University of MaltaFollow

Publication Date

2017

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 can be thought of as a generalization of 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.

Creative Commons License

This work is licensed under a Creative Commons Attribution 4.0 License.

Recommended Citation

Caro, Yair; Lauri, Josef; and Zarb, Christina (2017) "Two short proofs of the Perfect Forest Theorem," Theory and Applications of Graphs: Vol. 4: Iss. 1, Article 4.
DOI: 10.20429/tag.2017.040104
Available at: https://digitalcommons.georgiasouthern.edu/tag/vol4/iss1/4