Random Graphs with Few Disjoint Cycles

The classical ErdA‘s-Posa 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,.i¾ .i¾ .,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 ErdA‘s-Posa theorem with the extra 'redundancy' property that B-v is still a blocker for all but at most k vertices v ∈ B.