# Equitable partition of planar graphs

## TL;DR

This paper proves that all planar graphs can be partitioned equitably into specific subgraphs with degeneracy or forest properties, advancing understanding of graph decompositions.

## Contribution

It establishes new equitable partition results for planar graphs into degenerate graphs and forests, expanding graph partition theory.

## Key findings

- Planar graphs admit equitable 2-partitions into 3-degenerate graphs.
- Planar graphs admit equitable 3-partitions into 2-degenerate graphs.
- Planar graphs admit equitable 3-partitions into two forests and one graph.

## Abstract

An equitable $k$-partition of a graph $G$ is a collection of induced subgraphs $(G[V_1],G[V_2],\ldots,G[V_k])$ of $G$ such that $(V_1,V_2,\ldots,V_k)$ is a partition of $V(G)$ and $-1\le |V_i|-|V_j|\le 1$ for all $1\le i<j\le k$. We prove that every planar graph admits an equitable $2$-partition into $3$-degenerate graphs, an equitable $3$-partition into $2$-degenerate graphs, and an equitable $3$-partition into two forests and one graph.