The Erdős–Pósa property for edge-disjoint immersions in 4-edge-connected graphs

A graph H is immersed in a graph G if the vertices of H are mapped to distinct vertices of G, and the edges of H are mapped to paths joining the corresponding pairs of vertices of G, in such a way that the paths are pairwise edge-disjoint. In this paper, we show that the Erdős–Pósa property holds for packing edge-disjoint K_t-immersions in 4-edge-connected graphs. More precisely, for positive integers k and t, there exists a constant f(k,t) such that a 4-edge-connected graph G has either k edge-disjoint K_t-immersions, or an edge subset F of size at most f(k,t) such that G−F has no K_t-immersion. The 4-edge-connectivity in this statement is best possible in the sense that 3-edge-connected graphs do not have the Erdős–Pósa property.