The Erd\"os-P\'osa property for clique minors in highly connected graphs

TL;DR

This paper establishes a bound on the connectivity of highly connected graphs that guarantees either multiple disjoint clique minors or a small vertex set whose removal eliminates all such minors, extending the Erdős-Pósa property.

Contribution

It proves the Erdős-Pósa property for clique minors in highly connected graphs with explicit bounds, showing the minimal connectivity required for this property.

Findings

Existence of a function f linking connectivity and clique minors

Bound on connectivity is tight for p > 4

Either k disjoint K_p minors or small vertex set to eliminate all minors

Abstract

We prove the existence of a function f: N^2 -> N such that for all p,k in N every (k(p-3) + 14p+14) - connected graph either has k disjoint K_p minors or contains a set of at most f(p,k) vertices whose deletion kills all its K_p minors. For fixed p > 4, the connectivity bound of about k(p-3) is smallest possible, up to an additive constant: if we assume less connectivity in terms of k, there will be no such function f.