Equitable partition of graphs into induced forests

TL;DR

This paper proves new results on equitable partitions of graphs into induced forests, showing that graphs with certain coloring or degeneracy properties can be partitioned into a small number of such forests, including planar graphs.

Contribution

It establishes that graphs with acyclic colorings or bounded degeneracy can be equitably partitioned into a limited number of induced forests, confirming a conjecture for planar graphs.

Findings

Graphs with acyclic coloring with at most k colors can be partitioned into k-1 induced forests.

d-degenerate graphs can be equitably partitioned into k induced forests for k ≥ 3^{d-1}.

Planar graphs have equitable partitions into k induced forests for sufficiently large k.

Abstract

An equitable partition of a graph $G$ is a partition of the vertex-set of $G$ such that the sizes of any two parts differ by at most one. We show that every graph with an acyclic coloring with at most $k$ colors can be equitably partitioned into $k-1$ induced forests. We also prove that for any integers $d\ge 1$ and $k\ge 3^{d-1}$, any $d$-degenerate graph can be equitably partitioned into $k$ induced forests. Each of these results implies the existence of a constant $c$ such that for any $k \ge c$, any planar graph has an equitable partition into $k$ induced forests. This was conjectured by Wu, Zhang, and Li in 2013.