Equitable partition of plane graphs with independent crossings into induced forests

TL;DR

This paper proves that plane graphs with independent crossings can be equitably partitioned into a small number of induced forests, with bounds improving under certain girth conditions.

Contribution

It establishes new bounds for equitable partitions of plane graphs with independent crossings into induced forests, depending on girth constraints.

Findings

Partition into 8 induced forests for general case

Reduced bounds for higher girth graphs (down to 3 forests)

Improved understanding of graph decompositions with crossing constraints

Abstract

The cluster of a crossing in a graph drawing in the plane is the set of the four end-vertices of its two crossed edges. Two crossings are independent if their clusters do not intersect. In this paper, we prove that every plane graph with independent crossings has an equitable partition into $m$ induced forests for any $m\geq 8$. Moreover, we decrease this lower bound 8 for $m$ to 6, 5, 4 and 3 if we additionally assume that the girth of the considering graph is at least 4, 5, 6 and 26, respectively.