Two short proofs of the Perfect Forest Theorem

TL;DR

This paper presents two concise, elementary proofs of the Perfect Forest Theorem, which guarantees the existence of a perfect forest in any connected graph of even order, and provides polynomial algorithms for finding such forests.

Contribution

The authors offer two new short proofs of the Perfect Forest Theorem using elementary graph theory, along with polynomial-time algorithms for constructing perfect forests.

Findings

Two short proofs using elementary graph theory

Polynomial-time algorithms for finding perfect forests

Extension of the theorem to connected graphs of even order

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.