Packing Topological Minors Half-Integrally

A family F of graphs has the Erdős-Pósa property if for every graph G, the maximum number of pairwise disjoint subgraphs isomorphic to members of F contained in G and the minimum size of a set of vertices of G hitting all such subgraphs are bounded by functions of each other. Robertson and Seymour proved that if F consists of H-minors for some fixed graph H, then the planarity of H is equivalent with the Erdős-Pósa property. Thomas conjectured that the planarity is no longer required if the requirement for those H-minors being pairwise disjoint is replaced with being half-integrally disjoint. A restatement of this conjecture is that for every graph H, the optimal solutions of the pair of integer programming problems about packing and covering H-minors can be bounded in terms of each other once the integral requirement is relaxed to be half-integral. In this paper, we prove that this half-integral version of Erdős-Pósa property holds with respect to the topological minor containment, which easily implies Thomas’ conjecture. Indeed, we prove an even stronger statement in which those subdivisions are rooted at any choice of prescribed subsets of vertices. Precisely, we prove that for every graph H, there exists a function f such that for every graph G, every sequence (Rv : v ∈ V (H)) of subsets of V (G) and every integer k, either there exist k subgraphs G1, G2, ..., Gk of G such that every vertex of G belongs to at most two of G1, ..., Gk and each Gi is isomorphic to a subdivision of H whose branch vertex corresponding to v belongs to Rv for each v ∈ V (H), or there exists a set Z ⊆ V (G) with size at most f(k) intersecting all subgraphs of G isomorphic to a subdivision of H whose branch vertex corresponding to v belongs to Rv for each v ∈ V (H). Applications of this theorem include generalizations of algorithmic meta-theorems and structure theorems forH-topological minor free (orH-minor free) graphs to graphs that do not half-integrally pack many H-topological minors (or H-minors). This material is based upon work supported by the National Science Foundation under Grant No. DMS1664593. Email:[email protected]. Partially supported by NSF under Grant No. DMS-1664593.