Erdős-Pósa Ratio in Graph Theory

• The Erdős-Pósa ratio is a measure that quantifies the integrality gap between disjoint subgraph packings and their minimum covering sets in host graphs.
• It establishes key bounds, such as Θ(klog k) for cycles, and guides the analysis of trade-offs in graph properties like minors and hypergraphs.
• The concept has significant implications for algorithmic approximability and structural extremal limits, influencing research in graph decompositions and VC-dimension-based methods.

The Erdős-Pósa ratio quantifies the gap between packing and covering for specified subgraph patterns in host graphs. It is defined for a graph family $\mathcal{F}$ and an integer $k\geq1$ as the minimum value $f(k)$ such that every graph $G$ either contains $k$ pairwise disjoint subgraphs from $\mathcal{F}$ (packing), or admits a hitting set of size at most $f(k)$ meeting every member of $\mathcal{F}$ (covering). The ratio $f(k)/k$ measures the maximal "integrality gap" between the packing and covering numbers, controlling both algorithmic approximability and structural extremal limits.

1. Definitions and Formal Framework

Let $\mathcal{F}$ be a family of subgraphs (cycles, minors, balls, etc.). For any graph $G$, define:

• Packing number $\nu_{\mathcal{F}}(G)$: the largest integer $k$ such that $G$ contains $k$ pairwise vertex-disjoint subgraphs each isomorphic to a member of $\mathcal{F}$.
• Covering number $\tau_{\mathcal{F}}(G)$: the minimum size of a vertex (or edge) set $X$ such that $G-X$ contains no subgraph from $\mathcal{F}$.

$\mathcal{F}$ has the Erdős-Pósa property if there exists a function $f_{\mathcal{F}}$ (the "gap function") such that whenever $\nu_{\mathcal{F}}(G)<k$, one has $\tau_{\mathcal{F}}(G)\le f_{\mathcal{F}}(k)$ (Raymond et al., 2016). The Erdős-Pósa ratio is typically defined as $f_{\mathcal{F}}(k)/k$.

2. Classical Results and Tight Bounds for Cycles

The original theorem of Erdős and Pósa states that for the family $\mathcal{C}$ of all cycles in undirected graphs,

$f_{\mathcal{C}}(k) = \Theta(k\log k)$

and both vertex and edge versions achieve this ratio asymptotically (Raymond et al., 2016). For long cycles (length $\ge \ell$), the optimal vertex-covering function is:

$f(\ell, k) = O(k\ell + k\log k)$

yielding a ratio

$R(k, \ell) = O(\ell + \log k)$

(Mousset et al., 2016, Fiorini et al., 2012, Bruhn et al., 2014). Lower bounds $\Omega(k\ell + k\log k)$ are achieved via complete graphs and high-girth expanders.

Subgraph Family	Packing-Covering Gap $f(k)$	Erdős-Pósa Ratio $f(k)/k$
All cycles	$\Theta(k\log k)$	$\Theta(\log k)$
Long cycles (vertex)	$O(k\ell + k\log k)$	$O(\ell + \log k)$
Long cycles (edge)	$O(k^2\log k + k\ell)$	$O(k\log k + \ell)$
Odd cycles in planar	$O(k)$	$O(1)$
Chordless cycles	$O(k^2\log k)$	$O(k\log k)$

3. Tree Minors, Planar Minors, and Generalizations

For minors and special families:

• For any tree $T$ with $t$ vertices, the tight bound is:

$\tau_T(G)\le t(k-1)$

so ratio $t-t/k$ is achieved, with limit $t$ as $k\to\infty$ (Dujmović et al., 2024).

• Forest minors (max component size $t'$): $\tau_F(G)\le t\,k - t'$.
• Planar minors (with cycle): best known bounds are $O(k\log k)$, matching vertex cycles in lower order (Aboulker et al., 2017, Raymond et al., 2016). For the wheel $W_t$, $f_{W_t}(k) = \Theta(k\log k)$ holds.
• For fixed planar $H$, conjecture holds: $f_H(k)=\Theta(k\log k)$ (Aboulker et al., 2017).

In these cases, the limiting Erdős-Pósa ratio for $H$ (with cycles) is $\Theta(\log k)$ except for tree minors, which achieve linear ratios $t$.

4. Parity and Length Restrictions

• Odd cycles in planar graphs: ratio at most 4; no planar graph achieves ratio above 2 (sharpness open) (Puhlmann et al., 28 Dec 2025).
• Chordless cycles: ratio $O(k\log k)$ for induced cycles of length 4 ("holes"); induced cycles of length $\ge \ell$ for $\ell\ge5$ fail the property (Kim et al., 2017).
• Even cycles, cycles modulo constraints, and induced odd cycles may lack any finite ratio (no property holds in general host graphs) (Raymond et al., 2016).

5. VC-Dimension, Balls, and Hypergraph Generalizations

Given a hypergraph $H=(V,E)$ (e.g., $k$-neighborhood balls in graph $G$), packing and hitting set transversality numbers correspond to $\nu(H)$ and $\tau(H)$. For graphs with bounded distance VC-dimension $d$:

• The bound is $\tau_\ell(G)\le F_d(\nu_\ell(G))$, where $F_d$ is double-exponential in $d$ and quasi-polynomial in $n$ (the packing number) (Bousquet et al., 2014).
• For planar graphs, explicit (though large) bounds exist (planar diameter $2\ell$ balls covered by $O(10^4)$ balls of radius $\ell$).
• For general classes, polynomial bounds hold only for bounded expansion (Bousquet et al., 2014); linear or near-linear ratio is conjectured for planar graphs but not yet proved.

6. Proof Techniques and Algorithmic Implications

• Tree-like decompositions: central to achieving tight bounds, especially for minor-closed families (Raymond et al., 2016). Large treewidth or tree-partition width forces large packings; master theorems tie width to gap via superadditivity and divide-and-conquer, yielding logarithmic factors in $k$.
• Girth-minor method: large girth and minimum degree force large minors or cycles, establishing lower bounds via structural graph arguments and reduction steps.
• VC-theory and Ramsey: for balls, Sauer-Shelah and Ramsey arguments govern asymptotic behavior; Matoušek's (p,q)-Erdős-Pósa theorems connect VC-dimension and covering number.

Algorithmically, most constructive proofs yield polynomial-time routines finding either packing or covering sets matching the theoretical bounds for fixed parameters. For parity, minor, and prescribed-vertex variants, approximation ratios are governed directly by the Erdős-Pósa ratio.

7. Current State, Open Problems, and Conjectures

• Tightness: For most families, bounds are optimal up to constants in $k$ and auxiliary parameters (cycle length, tree size). Improvement is sought for edge-Erdős-Pósa for long cycles, where the smallest $O(k\log k+k\ell)$ hitting set is conjectured (Batenburg et al., 2019).
• Linear ratios: Only trees, forests, and certain planar subcubic immersions achieve strictly linear ratios; for graphs with cycles or more complex minors, logarithmic factors are necessary.
• Parity and modular constraints: For cycles of fixed parity or modular length, property may fail without further graph structural assumptions.
• Hypergraph generalization: For classes with bounded distance VC-dimension, ratio is governed by VC-dimension–dependent functions, drastically increasing in general classes. The possibility of linear ratio for balls in planar graphs is an outstanding conjecture (Bousquet et al., 2014).
• Conjectures: Notably, for planar minors $H$ with cycles, $f_H(k) = \Theta(k\log k)$ is widely conjectured (Aboulker et al., 2017); Tuza's conjecture for triangles seeks factor 2 in $f_\triangle(k)$ (Raymond et al., 2016).

8. Summary Table: Bounds and Ratios

Family / Setting	Packing Number	Gap Function $f(k)$	Ratio $f(k)/k$
Cycles (vertex/edge)	$k$	$\Theta(k\log k)$	$\Theta(\log k)$
Long cycles (vertex)	$k$	$O(k\ell + k\log k)$	$O(\ell + \log k)$
Long cycles (edge, best known)	$k$	$O(k\ell\log(k\ell))$	$O(\ell\log(k\ell))$
Odd cycles (planar)	$k$	$\leq 4k$	$\leq 4$
Tree minors ($T$ on $t$ vertices)	$k$	$t(k-1)$	$t-t/k$
Planar minors (cycle $H$)	$k$	$O(k\log k)$	$O(\log k)$
Balls (distance VC-dim $d$)	$n$	$\exp(n^{O(1)}2^{O(d)})$	double-exponential in $d$
Chordless cycles	$k$	$O(k^2\log k)$	$O(k\log k)$

The Erdős-Pósa ratio is central to quantitative graph theory, controlling the tradeoff between packings and coverings for critical graph substructures, with implications across structural, extremal, approximation, and algorithmic domains. The sharp characterization of this ratio remains an active area, especially in extensions to classes governed by combinatorial or parity constraints and hypergraph generalizations.