Random Graphs with Few Disjoint Cycles

Abstract

The classical Erdős-Pósa 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 Erdős-Pósa theorem with the extra 'redundancy' property that B − v is still a blocker for all but at most k vertices v ∈ B.

Maximum degree in minor-closed classes of graphs

“…Besides those discussed above, examples show that it can be constant (forbidding a star) and it can be linear (forbidding two disjoint triangles). The last statement follows from [10], where it is shown that the class Ex(C 3 ∪ C 3 ) is asymptotically the same as the class of graphs G having a vertex v such that G − v is a forest.…”

Random graphs containing few disjoint excluded minors

“…For two sequences of reals and which are positive for n sufficiently large, we write if . The next result extends Theorem 1.2 of on forests. Theorem Let be a proper addable minor‐closed class of graphs, with growth constant γ ; and let k be a fixed positive integer.…”

“…The inclusion (1) is ‘nearly an equality’. To be more precise, it was shown in that as Thus consists of together with an exponentially smaller class of ‘exceptional’ graphs. A similar result holds for unlabelled graphs ; we consider only labelled graphs in this paper.…”

“…Recall that the number f ( k ) in the Erdős‐Pósa theorem must be of order . From (3) and Theorem 1.1 it follows that by removing just k vertices we can obtain: a forest from almost every graph with at most k disjoint cycles ; more generally, a graph without any cycles of length at least from almost every graph with at most k disjoint cycles of length at least (see also ); a collection of cacti (that is, a graph with each edge in at most one cycle) from almost every graph with at most k disjoint subdivisions of the diamond graph see Fig. below. In contrast, by Remark 5.7 below, almost none of the graphs in can be turned into a series‐parallel graph by removing one vertex.…”

“…See for numerical values for these limiting probabilities in the case when is the class of forests. We shall actually prove a detailed extension of this result, Theorem 6.1 below, concerning the limiting distribution of the unlabelled ‘fragment’ graph formed from the vertices not in the ‘giant’ component.…”