A unified Erdős-Pósa theorem for cycles in graphs labelled by multiple abelian groups

In 1965, Erdős and Pósa proved that there is an (approximate) duality between the maximum size of a packing of cycles and the minimum size of a vertex set hitting all cycles. Such a duality does not hold for odd cycles, and Dejter and Neumann-Lara asked in 1988 to find all pairs ${(\ell, z)}$ of integers where such a duality holds for the family of cycles of length $\ell$ modulo $z$. We characterise all such pairs, and we further generalise this characterisation to cycles in graphs labelled with a bounded number of abelian groups, whose values avoid a bounded number of elements of each group. This unifies almost all known types of cycles that admit such a duality, and it also provides new results. Moreover, we characterise the obstructions to such a duality in this setting, and thereby obtain an analogous characterisation for cycles in graphs embeddable on a fixed compact orientable surface.

💡 Research Summary

The paper presents a comprehensive unification of Erdős‑Pósa type dualities for cycles under a wide variety of constraints, by introducing a framework of graphs whose edges are labelled with elements from several abelian groups. A cycle is called “allowable” if, for each group, its total label avoids a prescribed finite forbidden set. This model simultaneously captures many previously studied families of constrained cycles, such as cycles of length ℓ modulo z, S‑cycles (cycles meeting a given vertex set), cycles of bounded length, and homology‑constrained cycles on surfaces.

The authors first answer a long‑standing question of Dejter and Neumann‑Lara concerning which pairs (ℓ, z) admit an Erdős‑Pósa duality for cycles of length ℓ modulo z. Theorem 1.1 gives a precise combinatorial characterisation: (i) if 2 divides z then ℓ must be a multiple of the smallest 2‑prime factor, and (ii) there must be no three distinct prime factors of z all of which fail to divide ℓ. When both conditions hold, there exists a function f(k) such that every graph either contains k vertex‑disjoint ℓ‑mod z cycles or a vertex set of size at most f(k) intersecting all such cycles.

Moving to the group‑labelled setting, the paper identifies two algebraic conditions on the set A of allowed group values that are both necessary and sufficient for an Erdős‑Pósa duality to hold. Condition (1) requires that for every a∈A the subgroup generated by 2a does not intersect A; condition (2) forbids a configuration of three elements a, b, c where none of the three pairwise generated subgroups intersect A while the triple generated subgroup does. Theorem 1.2 shows that if either condition fails, one can construct a graph (the “Escher wall” obstruction) that has no two vertex‑disjoint allowable cycles and yet requires arbitrarily many vertices to hit all allowable cycles. Theorem 1.3 proves the converse for the natural case where each group Γ_j is allowed to take any value except a fixed finite set Ω_j: under the two algebraic conditions, a universal function f_{m,ω}(k) exists (independent of the specific groups) guaranteeing the Erdős‑Pósa property for allowable cycles.

A central contribution is the structural description of all possible obstructions (Definition 3.2, Theorem 3.3). Each obstruction consists of a large wall together with collections of paths arranged in three possible configurations—crossing, nested, or in series—around the wall’s boundary. The arrangement of these path families determines whether a cycle can be allowable; the obstruction forces any allowable cycle to use at least one path from each family, and forces an odd number of crossing families or a non‑trivial mixture of series families. These graphs admit at most a half‑integral packing of size t (each vertex belongs to at most two cycles) but no integral packing larger than two, while their hitting sets can be made arbitrarily large by enlarging the wall. This generalises Reed’s earlier characterisation of Escher‑wall obstructions for odd cycles.

The authors also study the impact of embedding constraints. On a fixed compact orientable surface S, large Escher walls cannot be embedded, so condition (1) of Theorem 1.1 becomes unnecessary. Theorem 1.5 gives the corresponding characterisation for graphs embeddable on S, allowing ℓ to be a multiple of either the smallest 2‑prime factor or its complementary factor z/p₁. Moreover, the framework yields Erdős‑Pósa results for cycles whose ℤ₂‑homology class lies in a prescribed set of allowable values, extending recent half‑integral results to integral ones on surfaces.

The paper also contrasts its approach with that of Huynh, Joos, and Wollan, who considered edge‑orientation labelings rather than edge‑labelings, and discusses why the two models are not directly interchangeable but can encode many overlapping constraints.

In summary, the work unifies and extends a large body of Erdős‑Pósa type theorems for constrained cycles, provides a clean algebraic characterisation of when such dualities hold, and supplies a detailed structural theory of the only possible obstructions. Open problems include extending the characterisation to arbitrary (possibly infinite) allowed sets A without the finite‑forbidden‑set assumption, and exploring non‑abelian or orientation‑based labelings. The results have immediate implications for graph minors, topological graph theory, and algorithmic applications where packing and covering of constrained cycles are central.