Random graphs with few disjoint cycles

TL;DR

This paper investigates the structure of random graphs with limited disjoint cycles, showing most have small blockers of size k and exploring properties like connectivity and chromatic number.

Contribution

It demonstrates that nearly all such graphs possess a small, redundant blocker and analyzes their structural properties, extending Erdős-Pósa theorem insights.

Findings

Most graphs have a small blocker of size k

Blockers are often redundant, remaining effective after removing most vertices

Properties like connectivity and chromatic number are characterized

Abstract

The classical Erd\H{o}s-P\'{o}sa theorem states that for each positive integer k there is an f(k) such that, in each graph G which does not have k+1 disjoint cycles, there is a blocker of size at most f(k); that is, a set B of at most f(k) vertices such that G-B has no cycles. We show that, amongst all such graphs on vertex set {1,..,n}, all but an exponentially small proportion have a blocker of size k. We also give further properties of a random graph sampled uniformly from this class; concerning uniqueness of the blocker, connectivity, chromatic number and clique number. A key step in the proof of the main theorem is to show that there must be a blocker as in the Erd\H{o}s-P\'{o}sa theorem with the extra `redundancy' property that B-v is still a blocker for all but at most k vertices v in B.