Equitable vertex arboricity of planar graphs

TL;DR

This paper proves that certain classes of planar graphs can be equitably partitioned into subsets inducing forests, confirming parts of a conjecture related to equitable vertex arboricity.

Contribution

It establishes equitable vertex arboricity results for specific planar graphs, advancing understanding of graph partitioning and confirming a conjecture in the field.

Findings

Partitioning of vertices into forests for graphs with independent small cycles

Results hold for all t ≥ 3

Partially confirms Wu, Zhang, and Li's conjecture

Abstract

Let $G_1$ be a planar graph such that all cycles of length at most 4 are independent and let $G_2$ be a planar graph without 3-cycles and adjacent 4-cycles. It is proved that the set of vertices of $G_1$ and $G_2$ can be equitably partitioned into $t$ subsets for every $t\geq 3$ so that each subset induces a forest. These results partially confirm a conjecture of Wu, Zhang and Li.