A tight Erd\H{o}s-P\'osa function for wheel minors

TL;DR

This paper establishes a tight Erdős-Pósa type result for wheel minors in graphs, showing a logarithmic bound on the size of vertex sets needed to hit all such minors, which is optimal up to a constant.

Contribution

It proves a tight Erdős-Pósa function for wheel minors, extending the understanding of minor-closed properties and their hitting sets in graphs.

Findings

Existence of a constant c(t) for wheel minors

Either k disjoint wheel minors or a small hitting set

Bound is optimal up to the constant c

Abstract

Let $W_t$ denote the wheel on $t+1$ vertices. We prove that for every integer $t \geq 3$ there is a constant $c=c(t)$ such that for every integer $k\geq 1$ and every graph $G$, either $G$ has $k$ vertex-disjoint subgraphs each containing $W_t$ as minor, or there is a subset $X$ of at most $c k \log k$ vertices such that $G-X$ has no $W_t$ minor. This is best possible, up to the value of $c$. We conjecture that the result remains true more generally if we replace $W_t$ with any fixed planar graph $H$.