Recent techniques and results on the Erdős–Pósa property

Abstract

Several min-max relations in graph theory can be expressed in the framework of the Erdős-Pósa property. Typically, this property reveals a connection between packing and covering problems on graphs. We describe some recent techniques for proving this property that are related to tree-like decompositions. We also provide an unified presentation of the current state of the art on this topic.

K4-expansions have the edge-Erdős-Pósa property

“…If P is a cycle, however, then we see by inspecting the blueprints that P must contain a vertex that has degree 3 in M , which we already had excluded. Thus we have shown (18). Now let us define F ′ .…”

“…Finally, consider the case when H ′ is of s ′ -type. Again if M is a module as in (18), then the path P is either an s ′ -t ′ -path, a t ′ -w ′ -path, or an s ′ -w ′ -path that passes through t ′ (note that F does not intersect M ). Now, for a ∈ {s ′ , w ′ } if it is possible to separate a from t ′ in H ′ − F by at most k edges, then let F a be such a set, and otherwise set F a = ∅.…”

In absence of long chordless cycles, large tree-width becomes a local phenomenon

“…A classic theorem of Erdős and Pósa [7] asserts that for every integer k there is an integer r such that every graph either contains k disjoint cycles or a set of at most r vertices meeting every cycle. This result has been the starting point for an extensive line of research, see the survey by Raymond and Thilikos [15]. Let F , G be classes of graphs and ≤ a containment relation between graphs.…”

A tight Erdős–Pósa function for long cycles

“…For example, Kőnig's theorem can be stated as follows: every bipartite graph contains either k vertex-disjoint edges or a set of f (k) = k − 1 vertices meeting all the edges. The above result of Erdős and Pósa has spawned a long line of papers about the duality between packing and covering of different families of graphs, directed graphs, hypergraphs, rooted graphs, and other combinatorial objects (see a recent survey of Raymond and Thilikos [18] for more information).…”