# Packing and covering odd cycles in cubic plane graphs with small faces

## TL;DR

This paper establishes tight bounds on how to make certain cubic plane graphs bipartite by edge deletion, providing new insights into their structure and related extremal properties.

## Contribution

The paper introduces a tight bound on edge deletion needed to bipartition 3-connected cubic plane graphs with small faces, extending previous results.

## Key findings

- Bound of rom the abstract for bipartitioning
- Characterization of extremal graphs achieving the bound
- Lower bounds on maximum cut and independent set sizes

## Abstract

We show that any $3$-connected cubic plane graph on $n$ vertices, with all faces of size at most $6$, can be made bipartite by deleting no more than $\sqrt{(p+3t)n/5}$ edges, where $p$ and $t$ are the numbers of pentagonal and triangular faces, respectively. In particular, any such graph can be made bipartite by deleting at most $\sqrt{12n/5}$ edges. This bound is tight, and we characterise the extremal graphs. We deduce tight lower bounds on the size of a maximum cut and a maximum independent set for this class of graphs. This extends and sharpens the results of Faria, Klein and Stehlik [SIAM J. Discrete Math. 26 (2012) 1458-1469].