The Erdos-Pósa Property for Odd Cycles in Highly Connected Graphs

A family F of graphs is said to have the Erdős–Pósa property, if for every integer k there is an integer f(k,F) such that every graph G either contains k vertex disjoint subgraphs each isomorphic to a graph in F or a set C of at most f(k,F) vertices such that G−C has no subgraph isomorphic to a graph in F . The term Erdős–Pósa property arose because in [3] Erdős and Pósa proved that the family of cycles has this property. The family of odd cycles does not have the Erdős–Pósa property, as we now show. For a graph G an odd cycle cover is a set of vertices C⊆V (G) such that G−C is bipartite. An elementary wall of height eight is depicted in Figure 1. An elementary wall of height h for h≥3 is similar. It consists of h levels each containing h bricks, where a brick is a cycle of length six. A wall of height h is obtained