Packing and Covering Immersions in 4-Edge-Connected Graphs

TL;DR

This paper establishes a new edge-variant Erdős-Pósa property for immersions in 4-edge-connected graphs, showing a duality between packing and covering such immersions with a function depending on connectivity.

Contribution

It proves the Erdős-Pósa property for immersions in 4-edge-connected graphs, demonstrating the optimal connectivity condition for this property.

Findings

Existence of a function f(k) for packing and covering immersions

The property holds specifically for 4-edge-connected graphs

The result is optimal, as it fails for 3-edge-connected graphs

Abstract

A graph $G$ contains another graph $H$ as an immersion if $H$ can be obtained from a subgraph of $G$ by splitting off edges and removing isolated vertices. In this paper, we prove an edge-variant of the Erd\H{o}s-P\'{o}sa property with respect to the immersion containment in 4-edge-connected graphs. More precisely, we prove that for every graph $H$, there exists a function $f$ such that for every 4-edge-connected graph $G$, either $G$ contains $k$ pairwise edge-disjoint subgraphs each containing $H$ as an immersion, or there exists a set of at most $f(k)$ edges of $G$ intersecting all such subgraphs. This theorem is best possible in the sense that the 4-edge-connectivity cannot be replaced by the 3-edge-connectivity.