Packing Topological Minors Half-Integrally

TL;DR

This paper proves that allowing half-integral packings of topological minors in graphs ensures a bounded relationship with coverings, confirming a conjecture and extending results to broader graph classes.

Contribution

It establishes the half-integral Erdős-Pósa property for packing and covering $H$-topological minors, confirming Thomas' conjecture and providing a framework for more general graph classes.

Findings

Half-integral packing bounds the number of $H$-topological minors.

Confirms Thomas' conjecture on half-integral packings.

Extends results to broader minor-closed graph classes.

Abstract

The packing problem and the covering problem are two of the most general questions in graph theory. The Erd\H{o}s-P\'{o}sa property characterizes the cases when the optimal solutions of these two problems are bounded by functions of each other. Robertson and Seymour proved that when packing and covering $H$-minors for any fixed graph $H$, the planarity of $H$ is equivalent to the Erd\H{o}s-P\'{o}sa property. Thomas conjectured that the planarity is no longer required if the solution of the packing problem is allowed to be half-integral. In this paper, we prove that this half-integral version of Erd\H{o}s-P\'{o}sa property holds for packing and covering $H$-topological minors, for any fixed graph $H$, which easily implies Thomas' conjecture. In fact, we prove an even stronger statement in which those topological minors are rooted at any choice of prescribed subsets of vertices. A number of results on $H$-topological minor free or $H$-minor free graphs have conclusions or requirements tied to properties of $H$. Classes of graphs that can half-integrally pack only a bounded number of $H$-topological minors or $H$-minors are more general topological minor-closed or minor-closed families whose minimal obstructions are more complicated than $H$. Our theorem provides a general machinery to extend those results to those more general classes of graphs without losing their tight connections to $H$.