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 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 two very short proofs of the Perfect Forest Theorem which use only elementary notions from graph theory. Both of our proofs yield polynomial time algorithms for finding a perfect forest in a connected graph of even order.