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The Erdős-Pósa Property for Colorful Minors
Evangelos Protopapas, Dimitrios M. Thilikos, Sebastian Wiederrecht
TL;DR
The paper asks which colorful graphs satisfy the Erdős–Pósa property under the colorful minor relation. It answers by proving equivalent structural, obstruction-based, and grid-like characterizations of exactly those graphs.
Problem
The paper asks which colorful graphs have the Erdős–Pósa property when annotated vertex constraints are represented through colors.
Method
The authors characterize crucial colorful graphs through obstruction minors and segregated grids, then prove cruciality equivalent to the Erdős–Pósa property.
Results
A colorful graph has the Erdős–Pósa property exactly when it is crucial, belongs to U, and excludes every member of O as a colorful minor.
Takeaways & Limitations
The characterization provides structural, obstruction-set, and grid-like descriptions of colorful graphs with the Erdős–Pósa property.
Takeaways & Limitations
The paper shows that no analogous characterization extends to finite sets of colorful graphs, even when all graphs use only one color.
Abstract
from arXiv · showhide
A colorful graph relation enhances the minor relation by merging color sets along contractions and by allowing the removal of colors; it generalizes rooted minors and models problems on graphs with several, possibly overlapping, annotated vertex sets. A graph has the Erdős-Pósa property for minors if and only if it is planar, by a classical theorem of Robertson and Seymour. In this work we determine, for the colorful minor relation, exactly which colorful graphs have the Erdős-Pósa property. Our characterization takes three equivalent forms. The first is structural: the colorful graphs with the property are those that can be drawn with all their colored vertices on one face and whose colors are, in a precise sense, laid out along that face without interleaving. The second is given by an obstruction set: they are those excluding every member of an explicit infinite family $\mathcal{O},$ of which only $\mathbf{O}(|I|^{4})$ members have colors that are a subset of $I,$ for every finite set $I$ of colors. The third is grid-like: they are exactly the colorful minors of unions of particular families of segregated grids, the colorful analogues of the grids that drive the classical proof.
1 Introduction
The paper studies the Erdős–Pósa relationship between packing and covering colorful minor models. It gives a complete characterization of colorful graphs with this property, extending the planar characterization for graphs without colors.
- Packing and covering: The Erdős–Pósa property asks whether a bounded packing number guarantees a bounded vertex covering number through a function of the packing size.A covering meets every relevant colorful minor model, while a packing consists of pairwise vertex-disjoint models.
- Colorful minors: Colorful minors extend ordinary minors by allowing edge and vertex deletion, color removal, and contractions that merge endpoint palettes.The relation is designed for graphs whose vertices carry finite sets of positive-integer colors.
- Main contribution: The paper gives a complete characterization of colorful graphs with the Erdős–Pósa property, extending the result that uncolored graphs have the property exactly when they are planar.The introduction emphasizes that colors produce subtle changes to the boundary of the characterization.
2 Basic definitions and statement of the main result
The section defines domains of colorful graphs closed under colorful minors and states a complete characterization of the nonempty colorful graphs with the Erdős–Pósa property. It also introduces segregated grids and obstruction families used in equivalent formulations and structural tools.
- Basic definitions: A domain is a class of colorful graphs closed under colorful minors, and restricting its palette to any finite color set preserves this property.For finite I, Z|I contains members whose palettes are subsets of I.
- Segregated grids: An (I, k)-segregated grid is a (qk × qk)-grid whose first-column vertices are divided into q consecutive color blocks, with all other vertices uncolored.The (∅, k)-segregated grid is the uncolored (k × k)-grid, and there are ⌈|I|!/2⌉ non-isomorphic (I, k)-segregated grids.
- Obstructions: For q colors, each obstruction family O_I is nonempty and finite, with size growing polynomially in q.Here I is a finite color set and q := |I|; the family is formally defined later in the paper.
- Main result: Theorem 2.1 gives four equivalent conditions: having the Erdős–Pósa property, being crucial, belonging to U, and excluding every member of O as a colorful minor.The theorem applies to every colorful graph with at least one vertex.
- Consequences: The class of colorful graphs with at least one vertex that have the Erdős–Pósa property is closed under colorful minors.This is stated as Corollary 2.2 and follows from the equivalence between the property and being crucial.
3 Crucial colorful graphs and their obstructions
Section 3 constructs crucial colorful graphs through progressively refined structural conditions and identifies their complete obstruction family. It proves that crucial graphs are exactly the colorful minors represented by the domain U and by sufficiently large segregated grids.
- Structural conditions: Color-facial graphs are characterized by excluding O0 ∪ O1, while color-segmented graphs are characterized by excluding O2.O0 consists of the two uncolored Kuratowski graphs, and the color-facial property is closed under colorful minors.
- Structural conditions: Component-wise bicolored graphs exclude O3, and single-component bicolored graphs exclude O4 as colorful minors.These conditions form part of the successive color restrictions used to define crucial colorful graphs.
- Obstruction family: The families ˜O1, ˜O2, O3, and O4, and therefore O, are infinite, but O restricted to any finite color set I is finite.For q=0,1,2,3,4,5,6, the respective sizes are 2, 6, 19, 42, 79, 137, and 226.
- Obstruction characterization: A colorful graph is crucial if and only if it excludes every member of the infinite obstruction family O as a colorful minor.The family O is the obstruction set of the domain Q of crucial colorful graphs.
- Segregated grids: Crucial colorful graphs are exactly the members of U, and every I-colorful member of U is a colorful minor of every sufficiently large (I,K)-segregated grid.For each such graph, a threshold K_W exists so that the colorful-minor relation holds for all K ≥ K_W.
4 The Erdős-Pósa property
Section 4 proves that every crucial colorful graph has the Erdős–Pósa property. The proof reduces hosts to bounded-torso-treewidth structures, applies a torso decomposition covering argument, and establishes the result first for connected crucial graphs before extending it to all crucial graphs.
- Color reduction: Color trimming preserves packings and coverings when colors outside the target palette are removed.For Q := ψ(H), packings and coverings of every size are identical in (G, χ) and (G, χ) ∩Q.
- Structural reduction: For every crucial colorful graph and packing order k, hosts either contain a packing of size k or admit a bounded-torso-treewidth reduction with restricted components.The reduction uses a constant c depending on the colorful graph and k; each component outside the selected set is restricted to the graph’s color set.
- Covering argument: Given bounded torso treewidth and restricted components, a standard tree-decomposition argument yields a bounded vertex set that blocks the colorful minor or isolates restricted components, unless a size-k packing exists.The covering set has size at most (t + 1)(k −1).
- Main result: Every crucial colorful graph has the Erdős–Pósa property.The connected case is established first, followed by the general case.
- Assembly argument: The final assembly argument constructs pairwise vertex-disjoint subgraphs containing the colorful minor, producing a packing and contradicting the assumption that no size-k packing exists.The argument uses disjointness claims and assembles components across the partition.
5 The lower bound
This section proves the lower-bound direction: every colorful graph with the Erdős–Pósa property is crucial. The proof constructs graph families with bounded packing but unbounded covering for each way cruciality can fail, using scarce surface resources.
- Proof strategy: Non-cruciality bounds packings while multiplication makes covering numbers grow without bound.The proof exhibits families containing 2k copies whose packing number stays bounded, while a representative construction satisfies cover_F,φ(G_2k, χ_2k) ≥ k.
- Proof strategy: The four failures of cruciality force copies to compete for scarce resources: Euler genus, disk boundary capacity, or linkage-crossing width.For color-facial cases, a disk supports only linearly many pairwise non-crossing objects reaching prescribed boundary stretches; non-single-component-bicolored patterns additionally require interleaving differently colored linkages.
- Obstruction transfer: The argument works with an obstruction colorful minor because packing transfers upward, but covering does not provide the needed lower bound.If (Z, ζ) ≤ (H, ψ), then pack_H,ψ(G, χ) ≤ pack_Z,ζ(G, χ) and cover_H,ψ(G, χ) ≤ cover_Z,ζ(G, χ).
- Main theorem: Every colorful graph with the Erdős–Pósa property is crucial.Equivalently, every non-crucial colorful graph fails the Erdős–Pósa property.
- Color-facial cases: For the color-facial cases, each realization meets at least three color intervals, yielding ℓ ≤ 2q_h and hence a bounded packing.The resulting bound is pack_H,ψ(H∘2k, ψ∘2k) ≤ ℓ ≤ 2q_h for every k.
6 Beyond a single colorful graph
For finite sets of colorful graphs, the Erdős-Pósa property is not characterized by the presence of a crucial member, even when all graphs share one palette. The section shows that interactions between members determine the outcome and leaves characterization of such finite sets open.
- A finite-set obstruction: Proposition 6.1 gives a finite set F of colorful graphs, all with palette {1}, that contains a crucial graph but lacks the Erdős-Pósa property.The set consists of F1 := (K3, ρ{1}) and F2 := (K1, ρ{1}) ⊔ (K5, ρ∅).
- A finite-set obstruction: The counterexample combines isolated color-1 vertices with uncolored graphs having bounded K5 packings but arbitrarily large K5 covers.For each n, the construction uses Hn with packK5,ρ∅(Hn) ≤ k0 and coverK5,ρ∅(Hn) ≥ n, alongside n isolated vertices of color 1.
- Relational behavior: The pair examples show that the property can change when one edge is added: {A, B1} satisfies it, whereas {A, B2} does not.B1 contains a rainbow C4 and B2 the corresponding rainbow P4; the edge closing the path into a cycle enables the colored part of A as a colorful minor.
- Relational behavior: Replacing a member by a colorful minor can destroy the property, so no condition examining each member separately can characterize finite sets.B2 is a colorful minor of B1, yet the two pairs lie on opposite sides of the Erdős-Pósa boundary.
- Open problem: Problem 6.2 asks for a characterization of finite sets of colorful graphs with the Erdős-Pósa property, with disconnected members identified as the first open case.The disconnected-member case is already open for pairs of annotated graphs.
Declaration on the use of generative AI
The authors used Claude for LaTeX editing, consistency and cross-reference checking, machine verification of Section 3 counting claims, and final TikZ figures. They state that the results, proofs, and constructions remain their own and that they reviewed assisted text.
- Declaration on the use of generative AI: Claude assisted with LaTeX editing, consistency and cross-reference checking, independent verification of Section 3 counting claims, and final TikZ figures.The authors specify these uses during preparation of the work.
- Declaration on the use of generative AI: The authors state that the paper’s results, proofs, and underlying constructions are their own.This declaration distinguishes tool assistance from authorship of the research contributions.
- Declaration on the use of generative AI: All tool-assisted text was reviewed and edited by the authors, who accept full responsibility.The passage reports author oversight and responsibility for the assisted material.