Note on Perfect Forests

TL;DR

This paper discusses the conditions under which a connected graph contains a perfect forest, characterized by all vertices having odd degrees within the forest and each tree being an induced subgraph, proving a theorem by A.D. Scott.

Contribution

It provides a concise proof of Scott's theorem linking perfect forests in connected graphs to even vertex counts.

Findings

A connected graph contains a perfect forest if and only if it has an even number of vertices.

The paper offers a simplified proof of Scott's theorem.

It clarifies the structural conditions for perfect forests in graphs.

Abstract

A spanning subgraph $F$ of a graph $G$ is called perfect if $F$ is a forest, the degree $d_F(x)$ of each vertex $x$ in $F$ is odd, and each tree of $F$ is an induced subgraph of $G$. We provide a short proof of the following theorem of A.D. Scott (Graphs & Combin., 2001): a connected graph $G$ contains a perfect forest if and only if $G$ has an even number of vertices.