Erdős-Pósa property of cycles that are far apart

We prove that there exist functions $f,g:\mathbb{N}\to\mathbb{N}$ such that for all nonnegative integers $k$ and $d$, for every graph $G$, either $G$ contains $k$ cycles such that vertices of different cycles have distance greater than $d$ in $G$, or there exists a subset $X$ of vertices of $G$ with $|X|\leq f(k)$ such that $G-B_G(X,g(d))$ is a forest, where $B_G(X,r)$ denotes the set of vertices of $G$ having distance at most $r$ from a vertex of $X$.

💡 Research Summary

The paper establishes a new distance‑restricted version of the classic Erdős–Pósa theorem. For any non‑negative integers k and d, the authors prove the existence of two functions f, g such that every graph G satisfies one of the following alternatives: (i) G contains k cycles whose pairwise vertex‑distance exceeds d (i.e., a d‑packing of k cycles), or (ii) there is a vertex set X with |X| ≤ f(k) such that after deleting the g(d)‑neighbourhood of X—formally B_G(X, g(d))—the remaining graph is a forest. In other words, a small “hitting” set together with a bounded‑radius “buffer” around it destroys all cycles when the graph does not admit k well‑separated cycles.

The authors give explicit bounds: f(k) = O(k¹⁸·polylog k) and g(d) = 19d. The proof blends several deep tools from structural graph theory:

1. Initial Packing Construction – They first extract a maximal 2d‑packing of cycles that are d‑unicyclic (each such cycle’s 2d‑ball contains no other cycle). Since a d‑packing of k cycles does not exist, the number p of these cycles is < k. Simultaneously they build a maximal d‑packing of “short” cycles (length ≤ 6d + 2), obtaining at most q < k such cycles.

2. Lemma 4 (Local Replacement) – Any cycle in G either lies within distance 2d of a d‑unicyclic cycle or within distance 3d of a short cycle. This guarantees that every cycle is captured by the balls around the previously selected cycles.

3. BFS‑Unicycle Framework – For each d‑unicyclic cycle C_i, they construct a C_i‑rooted spanning BFS‑unicycle U_i inside the 6d‑ball around C_i. This subgraph is unicyclic, contains C_i, and preserves distances to C_i. Edges outside U_i generate either a new cycle (when added to U_i) or a “lollipop” path P_e connecting two vertices of C_i.

4. Lemma 6 (Two‑Case Dichotomy) – Considering the collection of paths P_e, they form an auxiliary graph where two edges are adjacent if the corresponding paths are within distance d. If a large independent set exists, Simonovits’s theorem (which guarantees k vertex‑disjoint cycles in a graph of maximum degree 3 with many degree‑3 vertices) yields a d‑packing of k cycles. Otherwise, a small vertex set Y_i covers one endpoint of each external edge, and a bounded‑size set X_i covers all descendants of Y_i within distance 2r + d. This supplies the “buffer” around each C_i.

5. Combining Buffers – The authors define three bounded‑size sets: X₀ (one vertex from each short cycle), X₁ (the union of all Y_i buffers), and X₂ (a set that hits intersections of the 6d‑balls around different C_i’s). After removing the balls centered at X₀∪X₁∪X₂ with appropriate radii, the remaining graph decomposes into a forest F₀ (outside all 4d‑balls) and forests F_i (inside each 6d‑ball but outside the buffers).

6. Segment Analysis and Gyárfás–Lehel Helly‑type Theorem – Any remaining cycle must intersect F₀ and at least one F_i, producing “outside” and “inside” segments that are subpaths of the respective forests. Each segment’s d‑ball is a subtree of a tree. Applying the Gyárfás–Lehel theorem (a Helly‑type result for families of subtrees with bounded component count) yields either a small hitting set for all these d‑balls or a large collection of pairwise disjoint d‑balls. In the former case the hitting set becomes the final X; in the latter case, the disjoint balls give rise to many well‑separated segments, which again, via Simonovits’s theorem, produce a d‑packing of k cycles.

7. Quantitative Bounds – The authors carefully track the sizes of the constructed sets. Simonovits’s theorem requires at least s(k) ≈ 4k(log k + log log k + 4) degree‑3 vertices; the Gyárfás–Lehel bound ℓ*(k, 3) is shown to be O(k¹⁸). Consequently, the total size of the hitting set is bounded by O(k¹⁸·polylog k), giving the explicit f(k). The radius g(d) emerges from the accumulated additive constants (4d, 7d + 1, 13d, etc.) and is simplified to 19d.

The result resolves a conjecture posed by Chudnovsky and Seymour (a “coarse” Erdős–Pósa statement) and extends earlier work by Ahn, Gollin, Huynh, and Kwong, who handled the cases k = 2 or d = 1. By integrating Simonovits’s degree‑3 cycle packing technique with the Gyárfás–Lehel Helly‑type theorem, the authors create a robust framework for handling distance constraints in Erdős–Pósa‑type problems. The explicit bounds, while large, demonstrate that the property holds with polynomial dependence on k and linear dependence on d, opening avenues for further refinement (e.g., reducing the exponent 18) and for extending the approach to other graph families such as paths, minors, or topological substructures under distance constraints.