A unified half‐integral Erdős–Pósa theorem for cycles in graphs labelled by multiple abelian groups

Erdős and Pósa proved in 1965 that there is a 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 if we restrict to odd cycles. However, in 1999, Reed proved an analogue for odd cycles by relaxing packing to half‐integral packing. We prove a far‐reaching generalisation of the theorem of Reed; if the edges of a graph are labelled by finitely many abelian groups, then there is a duality between the maximum size of a half‐integral packing of cycles whose values avoid a fixed finite set for each abelian group and the minimum size of a vertex set hitting all such cycles. A multitude of natural properties of cycles can be encoded in this setting, for example, cycles of length at least ℓ$\ell$ , cycles of length p$p$ modulo q$q$ , cycles intersecting a prescribed set of vertices at least t$t$ times and cycles contained in given Z2$\mathbb {Z}_2$ ‐homology classes in a graph embedded on a fixed surface. Our main result allows us to prove a duality theorem for cycles satisfying a fixed set of finitely many such properties.


Introduction
A classical theorem of Erdős and Pósa [6] states that every graph contains either k vertex-disjoint cycles or a vertex set of size at most O(k log k) that hits all cycles of G.Such a theorem does not hold if we restrict to odd cycles; Lovász and Schrijver (see [26]) found a class of graphs having no two vertex-disjoint odd cycles and no small vertex set hitting all odd cycles.
In the setting of odd cycles, Reed [17] obtained an analogue of the theorem of Erdős and Pósa by relaxing the "vertex-disjoint" condition.A half-integral packing is a set of subgraphs such that no vertex is contained in more than two of them.Reed [17] proved that there is a function f such that every graph has a half-integral packing of at least k odd cycles or has a vertex set of size at most f (k) hitting all odd cycles.As an easy corollary of Reed's result, given a graph whose edges are labelled with Z 2 , there is a half-integral packing of at least k cycles, each of non-zero total weight, or a vertex set of size at most f (k) hitting all such cycles.
Very recently, Thomas and Yoo [25] extended this result to arbitrary abelian groups: they showed that there is a function f such that given a graph whose edges are labelled by an abelian group, there is a half-integral packing of at least k cycles each of non-zero total weight, or a vertex set of size at most f (k) hitting all such cycles.Kakimura and Kawarabayashi [12] proved a different kind of strengthening of the theorem of Reed.They showed that there is a function f such that every graph contains a half-integral packing of k odd cycles each of which intersects a prescribed set S of vertices or a vertex set of size at most f (k) hitting all such cycles.When S is the entire vertex set of the graph, this is equivalent to the theorem of Reed.This result can be encoded in the setting of graphs labelled with two abelian groups Z and Z 2 : in Z we label each edge incident with a vertex in S with 1, and all other edges with 0, and in Z 2 we label each edge with 1.The cycles which are of non-zero total weight with respect to both of these group labellings are precisely the cycles described by Kakimura and Kawarabayashi.Note that, since two groups are required for the encoding, this result is not covered by the previously mentioned result of Thomas and Yoo.
Our main theorem generalises all of these results to the setting of cycles in graphs labelled with a bounded number of abelian groups, whose values avoid a bounded number of elements of each group.For an abelian group Γ and a graph G, a function γ : E(G) → Γ is called a Γ-labelling of G.The γ-value of a subgraph H of G is the sum of γ(e) over all edges e in H.For an integer m, we write [m] for the set of positive integers at most m.We call a set of vertices which hits all subgraphs in a set H a hitting set for H. Theorem 1.1.For every pair of positive integers m and ω, there is a function f m,ω : N → N satisfying the following property.For each i ∈ [m], let Γ i be an abelian group, and let Ω i be a subset of Γ i .Let G be a graph and for each i ∈ [m], let γ i be a Γ i -labelling of G, and let O be the set of all cycles of G whose γ i -value is in , then for all k ∈ N there exists either a half-integral packing of k cycles in O, or a hitting set for O of size at most f m,ω (k).
We point out that the function f m,ω does not depend on the choice of groups Γ i or the subsets Ω i .For a graph labelled with a single abelian group Γ, Wollan [29] showed that if Γ has no element of order 2, then there are arbitrarily many vertex-disjoint cycles of non-zero γ-value, or a hitting set of bounded size for the γ-non-zero cycles.However, if Γ has an element of order 2, then as in the case of odd cycles, this does not hold [26,17].Hence, even when m = ω = 1, the half-integral condition cannot be removed in Theorem 1.1.
If the number m of given groups Γ i is unbounded or the size of Ω i is unbounded in Theorem 1.1, then such a function f m,ω does not exist, and moreover, for every integer n ≥ 2, a (1/n)-integral analogue of the Erdős-Pósa theorem does not hold.We discuss this in Section 3.
We now present some corollaries which help illustrate the power of this theorem.In the following corollaries, we use the function f m,ω in Theorem 1.1.As already mentioned, the cycles in a graph G which intersect a prescribed set of vertices can be encoded as precisely the non-zero cycles with respect to the Z-labelling which assigns value 1 to edges incident with vertices in S and 0 to all other edges.If instead each edge of G is assigned the integer that is the number of its endvertices which lie in S, then the cycles of total weight at least 2t are precisely the cycles which intersect the set S at least t times.Using multiple labellings with Z, we can encode the set of cycles which intersect each of a bounded number of sets at least t times each.Thus we obtain our first corollary.
Corollary 1.2.Let m and t be positive integers.For each i ∈ [m], let S i be a subset of the vertices of a graph G and t i ∈ [t], and let O be the set of all cycles of G containing at least t i vertices of S i for all i ∈ [m].Then for all k ∈ N there exists either a half-integral packing of k cycles in O, or a hitting set for O of size at most f m,t (k).
In fact, the construction above can be generalised, allowing us to convert a group labelling of the vertices of a graph to a group labelling of its edges.For an abelian group Γ and a graph G, a Γ-vertex-labelling of G is a function γ : V (G) → Γ, and the γ-value of a subgraph H of G is the sum of γ(v) over all vertices in H.In Section 3, we will discuss in detail how such a conversion works in general, and thus obtain the following corollary.
Corollary 1.3.Let m and ω be positive integers.For each i ∈ [m], let Γ i be an abelian group and let Ω i be a subset of Γ i .Let G be a graph, and for each i ∈ [m], let γ i be a Γ i -vertex-labelling of G, and let O be the set of all cycles of G whose γ i -value is in , then for all k ∈ N there exists either a half-integral packing of k cycles in O, or a hitting set for O of size at most f m,ω (k).
Our next corollary relates to graphs labelled with a fixed finite abelian group, where we obtain a similar result for the set of cycles of any specified value.In particular, this shows that for every pair of positive integers p and q, cycles of length p modulo q satisfy a half-integral analogue of the Erdős-Pósa theorem.Dejter and Neumann-Lara [4] showed that without the half-integral relaxation, the analogous Erdős-Pósa type result fails for cycles of length p modulo q whenever the least common multiple lcm(p, q) of p and q is divisible by 2p (see also [29]).When this condition is not met, Gollin, Hendrey, Kwon, Oum, and Yoo [8] show that the half-integral relaxation is required for an Erdős-Pósa type result in this setting if and only if lcm(p, q)/p is divisible by three distinct primes.
Corollary 1.4.Let Γ be a finite abelian group and let g ∈ Γ.Let G be a graph, and γ be a Γlabelling of G, and let O be the set of all cycles of γ-value g.Then for all k ∈ N there exists either a half-integral packing of k cycles in O, or a hitting set for O of size at most f 1,|Γ|−1 (k).
Our next corollary relates to graphs embedded on a fixed compact surface, where we obtain a similar result for the set of cycles contained in any given set of (first) Z 2 -homology classes.Huynh, Joos, and Wollan [10] proved that for graphs embedded on a fixed surface, cycles not homologous to zero in the Z-homology group satisfy a half-integral analogue of the Erdős-Pósa theorem.They used a different type of graph labelling, called a directed Γ-labelling.With (undirected) Γ-labellings, we can do the same thing with respect to Z 2 -homology classes.Since a compact surface has a finite abelian group as its Z 2 -homology group, we can obtain a half-integral analogue of the Erdős-Pósa theorem for cycles contained in any given set of Z 2 -homology classes.We discuss this further in Section 3. One nice feature of our main theorem is that it allows us to combine various properties of cycles together to obtain new results, as long as we take a bounded number of properties and can encode each of them with a bounded number of group labellings.Thus, we could combine any subset of these corollaries together and obtain a result of the same form.
Huynh, Joos, and Wollan [10] obtained a result similar to our main theorem for graphs with two directed group labellings, where the value of an edge is inverted if it is traversed in the reverse direction.They showed that a half-integral analogue of the Erdős-Pósa theorem holds for cycles whose values are non-zero in each coordinate.They conjectured that their result can be extended to graphs with more than two directed labellings.Because the Γ-labellings in this paper are equivalent to directed Γ-labellings when all elements of Γ have order 2, Theorem 1.1 implies that the conjecture of Huynh, Joos, and Wollan hold for graphs with a fixed number of directed labellings with such groups.The conjecture otherwise remains open, although there is a large overlap between the motivation of the conjecture and the consequences of our main theorem.We discuss directed group labellings in more detail in Section 10.
The structure of this paper is as follows.In Section 2, we introduce preliminary concepts, and we give a high-level overview of the proof of our main theorem and present proofs of corollaries in Section 3. In Section 4, we define a packing function and provide its application.In Section 5, we define a concept of a clean wall which is well-behaved for each of the labellings γ i .In Section 6, we prove a key lemma to find many vertex-disjoint paths attached to a wall.In Section 7, we prove useful lemmas on a product of abelian groups that will be used in the last step of the main theorem.Section 8 discusses how to obtain a desired cycle from a wall together with attached disjoint paths.We prove our main result in Section 9, and we discuss some open problems in Section 10.

Preliminaries
In this paper, all graphs are undirected simple graphs having no loops and multiple edges.For every abelian group, we regard its operation as an additive operation and denote its zero by 0.Even though we work on simple graphs, all the results are extended to multigraphs; given a multigraph instance, we can take a subdivision to produce an equivalent simple graph with group value 0 on new edges.
For an integer m, we write [m] for the set of positive integers at most m.
Let G be a graph.We denote by V (G) and E(G) the vertex set and the edge set of G, respectively.For a vertex set A of G, we denote by G − A the graph obtained from G by deleting all the vertices in A and all edges incident with vertices in A, and denote by For an edge e of G, we denote by G − e the graph obtained by deleting e.For two graphs G and H, let For a set G of graphs, we denote by G the union of the graphs in G.
For an integer t, a graph G is t-connected if it has more than t vertices and G − S is connected for all vertex sets S with |S| < t.
Subdividing an edge uv in a graph G is an operation that yields a graph containing one new vertex w, and with an edge set replacing uv by two new edges, uw and wv.A graph H is a subdivision of a graph G if H can be obtained from G by subdividing edges repeatedly.
Let A and B be vertex sets of G.An (A, B)-path is a path from a vertex in A to a vertex in B such that all internal vertices are not contained in A ∪ B. We also denote an (A, A)-path as an A-path.For a subgraph H of G, we shortly write as an H-path for a V (H)-path.A path is A-intersecting if it contains a vertex of A.
For a graph G, we denote by V =2 (G) the set of all vertices of G whose degrees are not equal to 2. A corridor of a graph G is a V =2 (G)-path of length at least 1.
Remark 2.1.Every graph in which no block is a cycle is the edge-disjoint union of its corridors.
by deleting both degree 1 vertices.Figure 1.An elementary (5, 7)-wall W depicted on the left with a 3-column-slice, and depicted on the right with a (4, 5)-subwall that is V =2 (W )-anchored.
A (c, r)-wall is a subdivision W of the elementary (c, r)-wall.If W is a (c, r)-wall for some suitable integers c and r, then we say W is a wall of order min{c, r}.We call a branch vertex corresponding to the vertex (i, j) of the elementary wall a nail of W , and denote by N W the set of nails of W . Remark 2.2.Any wall is a subdivision of a 3-connected planar graph.
For a subgraph H of the elementary wall, we denote by H W the subgraph of W corresponding to a subdivision of H.We call R W j or C W i the j-th row or i-th column of W , respectively.A subgraph W ′ of a wall W that is itself a wall is called a subwall of W .For a set S of vertices, we say a wall W is S-anchored if N W ⊆ S. We observe the following.
For an integer c ≥ 3, we call a subwall W ′ of a wall W a c-column-slice of W if the set of nails of W ′ is exactly N W ∩ V (W ′ ), there is a column of W ′ which is a column of W , and W ′ has exactly c columns, see Figure 1 for an example.Similarly, for an integer r ≥ 3, we call a subwall W ′ of a wall W an r-row-slice of W if the set of nails of W ′ is exactly N W ∩ V (W ′ ), there is a row of W ′ which is a row of W , and W ′ has exactly r rows.Note that in an r-row-slice W ′ of W , depending on the location, the first column of W ′ may be in the last column of W by the definition of a wall.See Figure 2 for an illustration.Given a wall W , a W -handle is a non-trivial W -path whose endvertices are degree-2 nails of W contained in the union of the first and last column of W .
Let W be a (c, r)-wall, let W ′ be a c ′ -column-slice of W for some 3 ≤ c ′ ≤ c, and let P be a W -handle.We define the row-extension of P to W ′ in W as the unique non-trivial W ′ -path in P ∪ {R W i : i ∈ [r]}.Note that such a P is a W ′ -handle.For a set P of vertex-disjoint Whandles, we define the row-extension of P to W ′ in W to be the set of row-extensions of the paths in P to W ′ in W .Note that these W ′ -handles are vertex-disjoint.
2.2.Linkages, separations, and tangles.Let G be a graph.For vertex sets A and B in G, a set P of vertex-disjoint (A, B)-paths of G is called a linkage from A to B, and its order is defined to be Its order is defined to be |A ∩ B|.We will use Menger's theorem.Theorem 2.4 (Menger [14]).Let A and B be vertex sets in a graph G, and k be a positive integer.Then G contains either a linkage of order k from A to B, or a separation (A ′ , B ′ ) of order less than k such that A ⊆ A ′ and B ⊆ B ′ .
We need a concept of a large wall dominated by a tangle, see [20, (2.3)].For a positive integer t, a set T of separations of order less than t is a tangle of order t in G if it satisfies the following.
(1) If (A, B) is a separation of G of order less than t, then T contains exactly one of (A, B) and (B, A).
Let W be a wall of order g with g ≥ 3 in a graph G. Let T W be the set of all separations (A, B) of G of order less than g such that G[B] contains a row of W .By the following simple lemma, we may replace the row with the column.The proof in [19] is for the grid but one can easily modify it for the wall.Lemma 2.5 (Robertson and Seymour [19,(7.1)]).Let W be a wall of order g in a graph G. Let (A, B) be a separation of order less than g.Then G[B] contains a row of W if and only if it contains a column of W .
Kleitman and Saks (see [19, (7.3)]) showed that T W is a tangle of order g.A tangle T in G dominates the wall W if T W ⊆ T .
Theorem 2.6 (Robertson, Seymour, and Thomas [20]).There exists a function f 2.6 : N → N such that if g ≥ 3 is an integer and T is a tangle in a graph G of order at least f 2.6 (g), then T dominates a (g, g)-wall W in G.
We will show that if a tangle dominates a wall W , then it also dominates every large subwall of W .We first prove the following lemma.
Lemma 2.7.Let W be a wall in a graph G and let S be a subset of V (G) of size exactly t.For each column C W x and row R W y of W , there are no more than t 2 nails of W which belong to components of W − S that do not intersect C W x ∪ R W y .Proof.We proceed by induction on t.The statement is trivial if t = 0. We may assume that G = W .Let S =: . Suppose there is a vertex v in S \ N W , and let P be the N W -path in W containing v. If both or neither of the endvertices of P are in components of W − S that intersect T , then we may apply the inductive hypothesis to S \ {v}.Otherwise, replacing v in S with the unique endvertex of P which is in a component of W − S that intersects T does not decrease the number of nails in components of W − S that do not intersect T .Hence, we may assume that S ⊆ N W .
For each i ∈ [t], let c(i) be the integer such that s i is in C W c(i) and let r(i) be the integer such that s i is in R W r(i) .For each i ∈ [t], let S c i be the set of nails v of W in C W c(i) − S such that the (v, R W y )-subpath of C W c(i) contains the vertex s i , and let S r i be the set of nails Let w ≥ t ≥ 3 be integers, let W be a wall of order w in a graph G, and let T be a tangle dominating W .If W ′ is a subwall of W of order t and In particular, if W ′ is N W -anchored, then T dominates W ′ .
Proof.Suppose for a contradiction that T does not dominate W ′ .Then G has a separation (A, B) of order less than t such that G[B] contains some row R ′ of W ′ and (A, B) / ∈ T .The order of T is at least w, so T contains (B, A) and therefore G[A] contains some row R of W .Let S := A ∩ B. Since W has more than |S| columns and R intersects each of them, G[A] contains some column C of W , and similarly G[B] contains some column C ′ of W ′ .By Lemma 2.7 applied to W , there are at most (t − 1) 2 nails of W in components of G − S which do not intersect R ∪ C. Similarly, there are at most (t − 1) 2.3.Groups.Let Γ i be a group for each i ∈ [m].We refer to the direct product of these groups by i∈[m] Γ i and denote by π j the projection map from i∈[m] Γ i to Γ j for j ∈ [m].For an element g of i∈[m] Γ i , we refer to the image π i (g) as the i-th coordinate of g.When we say Γ = i∈[m] Γ i is a product of groups, we implicitly use this notation.
For a non-empty set of elements S = {a i : i ∈ [t]} in a group Γ, we denote by S or a i : i ∈ [t] the subgroup generated by S, which is the intersection of all subgroups of Γ containing S.
2.4.Group-labelled graphs.Let Γ be an abelian group.A Γ-labelled graph is a pair of a graph G and a function γ : E(G) → Γ.We say that γ is a Γ-labelling of G.A subgraph of a Γ-labelled graph (G, γ) is a Γ-labelled graph (H, γ ′ ) such that H is a subgraph of G and γ ′ is the restriction of γ to E(H).By a slight abuse of notation, we may refer to this Γ-labelled graph by (H, γ).
For a Γ-labelled graph (G, γ) and a subgraph H ⊆ G, we define γ(H) as e∈E(H) γ(e), which we call the γ-value of H.Note that this definition implies that the γ-value of the empty subgraph is 0. We say that a subgraph H is γ-non-zero if γ(H) = 0, and otherwise, we call it γ-zero.
We will often consider the special case where Γ is the product i∈[m] Γ i of m abelian groups for a positive integer m.In this case, we denote by γ i the composition of γ with the projection to Γ We frequently take a subgroup Λ of Γ and consider a new labelling using the quotient group Γ/Λ.For a Γ-labelled graph (G, γ) and a subgroup Λ of Γ, the Γ/Λ-labelling λ defined by λ(e) := γ(e) + Λ for all edges e ∈ E(G) is the induced (Γ/Λ)-labelling of (G, γ).
We will use the following duality theorem between packing and covering of γ-non-zero A-paths.
Theorem 2.9 (Wollan [28]).Let k be a positive integer, let Γ be an abelian group, let (G, γ) be a Γ-labelled graph, and let A ⊆ V (G).Then G contains k vertex-disjoint γ-non-zero A-paths or there exists a set X ⊆ V (G) of size at most f 2.9 (k) := 50k 4 such that G − X has no γ-non-zero A-paths.
Let x be a vertex of G and let δ ∈ Γ be an element of order 2. For each edge e of G, let We say that γ ′ is obtained from γ by shifting by δ at x. Observe that this shift does not change the weight sum of a cycle because δ + δ = 0. We say two Γ-labellings γ 1 and γ 2 of G are shiftingequivalent if γ 1 can be obtained from γ 2 by a sequence of shifting operations.
The following lemma asserts that for γ-bipartite graphs we can find a shifting-equivalent Γlabelling γ ′ in which every corridor is γ ′ -zero.Similar ideas appear in Geelen and Gerards [7].
Lemma 2.10.Let Γ be an abelian group, let (G, γ) be a Γ-labelled graph and let H ⊆ G be a subdivision of a 3-connected graph Ĥ.If H is γ-bipartite, then γ is shifting-equivalent to a Γlabelling γ ′ such that every corridor of H is γ ′ -zero.
Proof.Let T be a spanning tree of Ĥ rooted at some r ∈ V ( Ĥ).It is enough to find a Γ-labelling γ ′ of G which is shifting-equivalent to γ, such that all corridors of H corresponding to edges in T are γ ′ -zero, because H is γ ′ -bipartite.Choose a Γ-labelling γ ′ shifting-equivalent to γ and a subtree T ′ of T containing r such that all corridors of H corresponding to edges in T ′ are γ ′ -zero, and subject to these conditions, |V (T ′ )| is maximised.
Suppose that T ′ = T .Then there is an edge vw of T such that v ∈ V (T ′ ) and w / ∈ V (T ′ ).Let P be the corridor of H corresponding to the edge vw.Since Ĥ is 3-connected, there is a cycle O in H − E(P ) containing v and w.Let O 1 and O 2 denote the distinct cycles in O ∪ P containing P .Since H is γ ′ -bipartite, we have that γ and hence γ ′ (P ) is an element of order at most 2. Let γ ′′ be a Γ-labelling of G obtained from γ ′ by shifting by γ ′ (P ) at w. Let T ′′ = T [V (T ′ ) ∪ {w}].Then all corridors of H corresponding to edges of T ′′ are γ ′′ -zero, contradicting the choice of γ ′ and T ′ .

Discussion
3.1.Proof Sketch.We now sketch the proof of Theorem 1.1, which will proceed by induction on k.We consider the group Γ := i∈[m] Γ i and a single Γ-labelling, which simplifies the arguments we present and in particular allows us to consider quotient groups.The goal will be to show that if the smallest hitting set T for the cycles in O is sufficiently larger than f m (k − 1), then there is a half-integral packing of k cycles in O.To construct this packing, in Section 4, we first find a tangle whose order is correlated with |T |/f m (k − 1), which allows us to construct a large wall as a subgraph in G.In particular, this wall will have the property that no cycle in O can be separated from the nails of the wall by deleting a small set of vertices.Our strategy will be to find disjoint sets of γ-non-zero paths and use the structure of the wall to connect them up to form cycles.But before we begin to do this, in Section 5 we find a subwall W of the original wall such that the N Wpaths in W have some nice homogeneity properties with respect to the group labelling.It would be simplest if we could guarantee that all N W -paths in W were γ-zero, but this is not feasible with arbitrary abelian groups.Instead, we deal separately with the factors Γ i of Γ for which we can guarantee that all N W -paths in W are γ i -zero, and the factors of Γ for which we cannot find any large subwall of W with this property.
Applying Theorem 2.9, we can find for every factor Γ i of Γ a large set of disjoint N W -paths which are γ i -non-zero.The difficulty here lies in combining these paths together such that for all i ∈ [m] the total γ i -value is not in Ω i .To achieve this, in Section 6 we show how to iteratively find sets of disjoint paths which are non-zero with respect to a quotient group defined in terms of the previously constructed paths, and link them up to the boundary of a subwall of W .In Section 7, we analyse conditions under which we can find a set of elements from a product of abelian groups whose sum avoids a finite set Ω i in each coordinate.In Section 8, we discuss how to combine the disjoint sets of paths into cycles using the wall.This allows us to find a half-integral packing of k cycles whose γ i -values are in Γ i \ Ω i for every factor Γ i of Γ for which the N W -paths in W are γ i -zero.
To deal with each remaining factor Γ i , we observe that since no large subwall of W is γ i -bipartite, every large subwall of W contains a γ i -non-zero cycle.In fact, we iteratively find disjoint cycles in W which are non-zero with respect to quotient groups defined in terms of the previously constructed cycles.We link these cycles up to the half-integral packing of k-cycles we have constructed, and by rerouting through them, transform each cycle in our half-integral packing into a cycle in O.

3.2.
Obstructions for integral and half-integral Erdős-Pósa type results.One fundamental obstruction to Erdős-Pósa type results, known as the Escher wall, is due to Lovász and Schrijver (see [26]).An Escher wall of height n is obtained from an (n, n)-wall W by adding a family (P 3 for an illustration.They observed that there are no two disjoint cycles in such an Escher wall which each contain an odd number of paths in (P i : i ∈ [n]), but for any set S of at most n − 1 vertices, there is a cycle of the Escher Wall containing exactly one path in (P i : i ∈ [n]).Using this construction, Thomassen [26] argued that an analogue of the Erdős-Pósa theorem does not hold for odd cycles.It can be further used to show that the same holds for cycles of length ℓ modulo m whenever m is an even positive integer and ℓ is an odd integer with 0 < ℓ < m, see Wollan [29].
Many previous half-integral Erdős-Pósa type results have relied on characterising Escher walls as the fundamental obstructions to finding integral packings of certain classes of cycles.Half-integral packings are then obtained using the structure of the Escher Wall.In our setting, the Escher Wall is not the only possible obstruction which can arise.In fact, unlike Escher walls, the following type of obstruction can occur even in planar graphs.For this reason, some of the structural results which have been used to obtain other half-integral Erdős-Pósa type results are unlikely to be useful in our setting.For example, we do not use the flat wall theorem of Robertson and Seymour [22,21] in this paper, and there is no obvious way to significantly simplify our proofs by doing so.Proposition 3.1.Let G be the n × n-grid, and let O be the set of cycles of G which contain at least one edge of the top row R t of G, at least one edge of the bottom row R b of G, and at least one edge from the leftmost column C ℓ of G, and let O ′ be the set of cycles of G which contain exactly one edge of the top row R t of G, exactly one edge of the bottom row R b of G, and exactly one edge from the leftmost column C ℓ of G. Then every pair of cycles in O intersect, but there is no hitting set for O ′ of size less than (n − 1)/2.
Proof.We may assume that n ≥ 3. Note that the graph G ′ obtained from G by adding a vertex z adjacent to every vertex of R t ∪ R b ∪ C ℓ is planar.Suppose for contradiction that there are disjoint cycles O 1 and O 2 in O, and let H be the component of to a vertex of H, and similarly a vertex v 2 in O 1 ∩ R t adjacent to a vertex of H and a vertex v 2 in O 1 ∩ R t adjacent to a vertex of H. Contracting C 1 to a triangle on {v 1 , v 2 , v 3 } and H to a single vertex, we find a K 5 -minor in G ′ , contradicting Wagner's Theorem.Now consider S ⊆ V (G) of size less than (n − 1)/2.Note that there are two adjacent columns C i and C i+1 which do not intersect S, and likewise two adjacent rows R j and R j+1 which do not intersect S. It is easy to see that the subgraph of G induced on the vertices in As an example, consider an n × n-grid in which all edges on the top row are subdivided exactly 14 times, all edges of the bottom row are subdivided exactly 69 times, all edges of the leftmost column are subdivided exactly 20 times, and all other edges are subdivided exactly 104 times.By Proposition 3.1, there are no two vertex-disjoint cycles of length 1 mod 105, and no hitting set for these cycles of size less than (n − 1)/2.
Another consequence of Proposition 3.1 is that an analogue of the Erdős-Pósa theorem does not hold for cycles intersecting three prescribed vertex sets.To see this, consider an n × n-grid, let S 1 be the set of all vertices on the top row, let S 2 be the set of all vertices of the bottom row, and let S 3 be the set of all vertices of the leftmost column.By Proposition 3.1, there are no two vertex-disjoint cycles each containing at least one vertex from each of S 1 , S 2 , S 3 , and no hitting set for these cycles of size less than (n − 1)/2.
Our main theorem demonstrates that it is not easy to find non-trivial obstructions for halfintegral Erdős-Pósa type results.However, the following proposition does allow us to describe settings in which even a half-integral Erdős-Pósa type result is not possible.Proposition 3.2.Let t and c be positive integers, let Γ be an abelian group and let Ω be a subset of Γ such that there is an element g ∈ Γ and an integer d > (c − 1)t such that d is the minimum integer greater than 2 for which dg / ∈ Ω.Then there is a graph G with Γ-labelling γ such that, for the set O of all cycles of G whose γ-values are not in Ω, every c cycles in O share a common vertex, but there is no hitting set for O of size less than t.
Proof.The c = 1 case is trivial, so we may assume c ≥ 2. Let n := ⌈cd/(c − 1)⌉ − 1, let G := K n , and let γ be the Γ-labelling assigning g to every edge of G.By construction, every cycle in O has length greater than (c − 1)n/c, and so every c cycles in O share a common vertex.However, every cycle of length d in G is in O, so the smallest hitting set for O has size n − (d − 1) > t.
As an example, consider an infinite abelian group which contains arbitrarily large finite cyclic subgroups.One consequence of Proposition 3.2 is that there is no half-integral Erdős-Pósa type result for cycles of weight zero in graphs labelled with such a group.
As another consequence, we obtain a lower bound on the functions mentioned in Theorem 1.1 which depends on both m and ω.

3.3.
Relating vertex-labellings to edge-labellings.We now demonstrate how to convert a group labelling of the vertices of a graph to a group labelling of its edges.Corollary 1.3 follows easily from Theorem 1.1 after applying the following lemma to each of the vertex-labellings it mentions.
Lemma 3.4.Let Γ be an abelian group, let Ω ⊆ Γ be a finite subset, let G be a graph, and let γ be a Γ-vertex-labelling of G. Then there is a group Γ ′ , a subset By the fundamental theorem of finitely generated abelian groups, there exist an integer m and an isomorphism ϕ from Γ ′′ to a product i∈[m] Γ i , where each Γ i is either Z, or a cyclic group , let e i denote the element of Γ ′′ such that π j (ϕ(e i )) = 1 if i = j, and π j (ϕ(e , we define a homomorphism ψ j from Γ ′′ to Γ ′ by setting ψ j (e i ) := je ′ i on the generators.Note that ψ 2 is injective since the kernel of ψ 2 is trivial, that the image of ψ 2 is 2Γ ′ , and that ψ 1 (2g) = ψ 2 (g) for all g ∈ Γ ′′ .We define Ω ′ := ψ 2 (Ω) and a Γ ′ -labelling γ ′ of G for an edge e = vw of G by setting γ ′ (e) = ψ 1 γ(v) + γ(w) .Note that for a cycle O of G, we have that Hence, the result follows from the injectivity of ψ 2 .
3.4.Graphs embedded on a surface.We now discuss how our result applies to graphs embedded on a surface, where we consider the first homology group with coefficients in Z 2 .Huynh, Joos, and Wollan [10, Proposition 5] demonstrated that given a graph G embedded on a surface whose Z-homology group is Γ, there is a directed Γ-labelling of G so that the set of cycles in G that are homologous to zero is exactly the set of cycles having group value 0 in the labelling.This allowed them to obtain for graphs embedded on a surface a half-integral Erdős-Pósa result for the non-null-homologous cycles of the embedding.Our result works in essentially the same way.
A graph H is called even if every vertex of H has even degree.For a graph G, let C(G) denote the cycle space of G over Z 2 , that is the vector space of all even subgraphs H of G with the symmetric difference as the operation.Proposition 3.5.Let G be a graph, let Γ be an abelian group, and let φ : C(G) → Γ be a group homomorphism.Then there is a Γ-labelling γ of G such that γ(H) = φ(H) for every even subgraph H of G.
Proof.Without loss of generality, we may assume that G is connected.Let T be a spanning tree of G.For each edge e ∈ E(G) \ E(T ), let C e,T denote the unique cycle in T + e.We define γ(e) := 0 for each e ∈ E(T ) and γ(e) := φ(C e,T ) for each e ∈ E(G) \ E(T ).The statement now trivially follows, because the set {C e,T : e ∈ E(G) \ E(T )} forms a basis of the cycle space (see [5, Theorem 1.9.5]).Now for a graph G embedded in a surface Σ, the map assigning each even subgraph its Z 2homology class is a group homomorphism from C(G) to the Z 2 -homology group of Σ.Hence, Corollary 1.5 follows with Proposition 3.5 from Theorem 1.1.
Note that for a closed orientable surface, the set of simple closed curves homologous to zero for the Z 2 -homology is exactly the same as for the Z-homology.This follows the universal coefficient theorem (see [9]), which allows us to relate the Z-homology with the Z 2 -homology by taking all coefficients modulo 2. We then apply a classical result which states that no simple closed curve has Z-homology class kh for any integer k ≥ 2 and any non-zero element h of the Z-homology (see for example [23]).Hence, in the case of graphs embedded on closed orientable surfaces, we recover the result of Huynh, Joos and Wollan for non-null-homologous cycles.

Packing functions and hitting sets
In this section, we introduce the concept of packing functions as a tool to generalise the ideas of both integral and half-integral packings of subgraphs, which enables us to discuss these and similar ideas in a unified way.
For a function ν from the set of subgraphs of a graph G to the set of non-negative integers, we say that ) whenever H and H ′ are vertex-disjoint subgraphs of G, and • ν is a packing function for G if it is monotone and additive.Now let ν be a packing function for a graph G.For a subgraph H ⊆ G, we say a set T ⊆ V (H) is an ν-hitting set for H if ν(H − T ) = 0. We define τ ν (H) as the size of a smallest ν-hitting set of H.Note that in the traditional sense of the word, a ν-hitting set of G is a hitting set for the minimal subgraphs H ⊆ G for which ν(H) ≥ 1.
For example, a function mapping a subgraph H of G to the maximum number of vertex-disjoint cycles in H is a packing function of G.
The following lemma argues that if ν(G) is small but G has no small ν-hitting set, then every minimum ν-hitting set induces a tangle of large order.Similar arguments for specific packing functions appear many times in the literature, see [10] and [18] for instance.Lemma 4.1.Let ν be a packing function for a graph G and let T ⊆ V (G) be a minimum ν-hitting set for G of size t.Let T T be the set of all separations (A, B) of G of order less than t/6 such that |B ∩ T | > 5t/6.If τ ν (H) ≤ t/12 whenever H is a subgraph of G with ν(H) < ν(G), then T T is a tangle of order ⌈t/6⌉.
Proof.First, we show the following claim.
Claim.Let X, Y ⊆ T be disjoint sets with |X| = |Y | ≥ t/6.Then there is a linkage in G from X to Y of order |X| containing no vertex in Z := T \ (X ∪ Y ).
Proof.Suppose for a contradiction that there is no such linkage.By Menger's theorem applied to G − Z, there is a separation (A, B) of G of order strictly less than  Let us now turn our attention to packing functions ν for a Γ-labelled graph (G, γ) for an abelian group Γ.The following lemma is useful for converting between γ-non-zero cycles and γ-non-zero paths.We will appeal to it in the final lemma of this section, and again in Lemma 6.3.Lemma 4.2.Let Γ be an abelian group, let (G, γ) be a Γ-labelled graph, let O be a γ-non-zero cycle in G, and let T ⊆ V (G).If G contains three vertex-disjoint (V (O), T )-paths P 1 , P 2 , P 3 , then H := O ∪ P 1 ∪ P 2 ∪ P 3 contains a γ-non-zero T -path.
Proof.We may assume that i be the two paths in H each having j∈ [3]\{i} V (P j ) ∩ T as its set of endvertices, where Hence, for some i ∈ [3], one of the paths And since this path is the edgedisjoint union of T -paths, it contains a γ-non-zero T -path, as desired.
Given an abelian group Γ and a Γ-labelled graph (G, γ), we are interested in the packing function ν for G which maps a subgraph H of G to the size of the largest half-integral packing of the type of cycles of H we are considering.To prove our main result, we will need the following tool for finding disjoint sets of paths in G which are non-zero with respect to the induced labelling of certain quotient groups, which we will later construct.Lemma 4.3.Let u, k be positive integers such that f 2.9 (k) < u − 2. Let Γ be an abelian group, let (G, γ) be a Γ-labelled graph, and let ν be a packing function for G such that • every minimal subgraph H of G with ν(H) ≥ 1 is a γ-non-zero cycle, • τ ν (H) ≤ 3u for every subgraph H of G with ν(H) < ν(G), and Let T ⊆ V (G) be a minimum ν-hitting set for G and let N ⊆ V (G) such that for every S ⊆ V (G) of size less than u, there is a component of G − S containing a vertex of N and at least 4u vertices of T .Then G contains k vertex-disjoint γ-non-zero N -paths.
Proof.Suppose that G does not contain k vertex-disjoint γ-non-zero N -paths.By Theorem 2.9, there exists , contradicting the assumption that T is a minimum ν-hitting set.

Clean walls
In the proof of our main theorem in Section 9, we will apply Theorem 2.6 and Lemma 4.1 to construct a wall W in a group-labelled graph.However, it will be useful to move to a large subwall of W which has some nice homogeneity properties.For this purpose, we introduce the following notion of cleanness.
Let Γ = i∈[m] Γ i be a product of m abelian groups and let (G, γ) be a Γ-labelled graph.Given a subset Z ⊆ [m] and an integer ℓ, we say that a wall W in G is (γ, Z, ℓ)-clean if (1) every N W -path in W is γ i -zero for all i ∈ Z, and (2) W has no (ℓ, ℓ)-subwall which is γ i -bipartite for all i ∈ [m] \ Z. Lemma 5.1.Let Γ = i∈[m] Γ i be a product of m abelian groups, let (G, γ) be a Γ-labelled graph, let ψ : {0} ∪ [m + 1] → N ≥3 be a function, and let W be a wall of order ψ(0) + 2 in G. Then there exist a Γ-labelling γ ′ of G shifting-equivalent to γ, a subset Z of [m], and a (γ ′ , Z, ψ(|Z| Among all Γ-labellings γ ′ of G shifting-equivalent to γ, we choose γ ′ maximising the number of elements i ∈ Z such that all corridors of W ′ are γ ′ i -zero.If there is i ∈ Z such that some corridor of W ′ is not γ ′ i -zero, then Lemma 2.10 applied to Γ i yields the Γ-labelling γ ′′ for which every corridor of W ′ is γ ′′ i -zero, thus contradicting the choice of γ ′ .Thus all corridors of W ′ are γ ′ i -zero for all i ∈ Z. In a sense, the notion of cleanness helps us to generalise the ideas of Thomassen [26] who proved the following result.Proposition 5.2 (Thomassen [26]).There exists a function w 5.2 : N 2 → N satisfying the following.Let t and w ≥ 3 be integers, let Γ be an abelian group generated by an element of order at most t, and let (W, γ) be a Γ-labelled wall of order w 5.2 (t, w).Then W contains a (w, w)-subwall W ′ such that γ(P ) = 0 for all corridors P of W ′ .
We extend Proposition 5.2 to a group generated by a fixed number of generators.
Lemma 5.3.There exists a function w 5.3 : N 3 → N satisfying the following.Let q, t and w ≥ 3 be integers, let Γ be an abelian group generated by q elements each of order at most t, and let (W, γ) be a Γ-labelled wall of order w 5.3 (q, t, w).Then W contains a (w, w)-subwall W ′ such that γ(P ) = 0 for all corridors P of W ′ .
Proof.We define • w 5.3 (1, t, w) := w 5.2 (t, w), and • w 5.3 (q, t, w) := w 5.2 (t, f 5.3 (q − 1, t, w)) for all integers q ≥ 2. We prove the lemma by induction on q with Proposition 5.2 as the base case.So let q ≥ 2, and let Γ = {x i : i ∈ [q]} for a suitable set of q generators each of order at most t.Let Γ 1 := x q and Γ 2 := {x i : i ∈ [q − 1]} , and let γ ′ be a Γ 1 -labelling of W such that for every edge e of W , we have γ(e) + γ ′ (e) ∈ Γ 2 .By Proposition 5.2, W has a (w 5.3 (q − 1, t, w), w 5.3 (q − 1, t, w))-subwall W ′ such that γ(P ) ∈ Γ 2 for all corridors P of W ′ .By the induction hypothesis, W ′ has a (w, w)subwall W ′′ such that γ(P ) = 0 for all corridors P of W ′′ .Note that as W ′ is a subwall of W , all corridors of W ′′ in W ′ are corridors of W ′′ in W .
The following variation allows us to take advantage of our notion of cleanness and will be needed for Lemma 8.1.
Corollary 5.4.There exists a function w 5.4 : N 3 → N satisfying the following.Let q, t and w ≥ 3 be integers, let Γ be an abelian group, and let Λ be a subgroup of Γ generated by q elements each of order at most t, and let (W, γ) be a Γ-labelled wall of order w 5.4 (q, t, w).If γ(O) ∈ Λ for all cycles O of W , then W contains a γ-bipartite (w, w)-subwall.
The wall W is a subdivision of some 3-connected planar graph Ĥ.Let T be a spanning tree of Ĥ, rooted at an arbitrary vertex r.Choose a Γ-labelling γ ′ shifting-equivalent to γ and a subtree T ′ of T containing r such that γ ′ (P ) ∈ Λ for all corridors P of W corresponding to edges in T ′ , and subject to these conditions, |V (T ′ )| is maximised.
Suppose that T ′ = T .Then there is an edge vw of T such that v ∈ V (T ′ ) and w / ∈ V (T ′ ).Let Q be the corridor of W corresponding to the edge vw.Since Ĥ is 3-connected, there is a Λ for all corridors P of W corresponding to edges of T ′′ , contradicting our choice of γ ′ and T ′ .Therefore T ′ = T .Now, observe that γ ′ (P ) ∈ Λ for every corridor P of W , because Λ ⊆ Λ and for every cycle O of W , we have γ ′ (O) = γ(O) ∈ Λ.Let γ ′′ be the Λ-labelling of W which assigns an arbitrary edge e P of each corridor P the value γ ′ (P ), and all other edges the value 0. By Lemma 5.3, there is a (w, w)-subwall W ′ of W which in particular is γ ′′ -bipartite, and hence γ ′ -bipartite.Since γ ′ and γ are shifting-equivalent, W ′ is γ-bipartite.

Handling handles
This section is dedicated to proving the following key lemma, which allows us to iteratively find sets of vertex-disjoint handles.Lemma 6.1.There exist functions w 6.1 : N 2 → N and f 6.1 : N → N satisfying the following.Let k, t and c be positive integers with c ≥ 3, let Γ be an abelian group, and let (G, γ) be a Γ-labelled graph.Let W be a wall in G of order at least w 6.1 (k, c) such that all corridors of W are γ-zero.
Before we can prove this lemma, we need to establish a variety of other lemmas.At the heart of the proof, we have the following natural result, which we will iteratively apply to decouple the sets of vertex-disjoint handles that we will construct.Huynh, Joos, and Wollan [10, Lemma 27] proved a somewhat similar result for oriented group-labelled graphs.Lemma 6.2.Let k, t be positive integers, let Γ be an abelian group, let (G, γ) be a Γ-labelled graph, and let T be a subset of V (G).For each i ∈ [t − 1], let P i be a set of T -paths of size 4k such that the paths in i∈[t−1] P i are vertex-disjoint.
If G contains k vertex-disjoint γ-non-zero T -paths, then there exist a set Q t of k vertex-disjoint γ-non-zero T -paths and a subset Q i ⊆ P i of size k for each i ∈ [t − 1] so that the paths in i∈[t] Q i are vertex-disjoint.
Proof.Let Q t be a set of k vertex-disjoint γ-non-zero T -paths such that the number of edges of paths in Q t that are not contained in any path in P := i∈[t−1] P i is as small as possible.For each j ∈ [t − 1], let P * j be the set of paths in P j that do not contain an endvertex of a path in Q t .We have . Let P * * j be the set of all paths in P * j intersecting a path in Q t .Assume that for some j ∈ [t − 1] we have |P * * j | ≥ k + 1.Then there are two paths P 1 , P 2 ∈ P * * j such that when traversing from an endvertex p i of P i for each i ∈ [2], P 1 and P 2 first meet the same path Q ∈ Q t .For each i ∈ [2], let q i be the first intersection of P i and Q when traversing P i from p i .Let v and w be the endvertices of Q such that the distance between v and q 1 in Q is smaller than the distance between v and q 2 in Q.Let Q 1 , Q 2 , Q 3 be the subpaths of Q from v to q 1 , from q 1 to q 2 , and from q 2 to w, respectively.Also, for each i ∈ [2], let P ′ i be the subpath of P i from p i to q i , see Figure 4. Since the paths of i∈[t−1] P i are vertex-disjoint and v, w / ∈ V (P 1 ∪ P 2 ), both Q 1 and Q 3 contain edges not in a path of i∈[t−1] P i ; for instance, edges incident with q 1 or q 2 .
By assumption, 2 such that R = Q and γ(R) = 0, then by replacing Q with R, the number of edges of paths in Q t that are not contained in any path in i∈[t−1] P i decreases.Therefore, by the assumption on Q t , we have γ(R) = 0 for every such path R. It implies that (1) Segments of the paths P 1 , P 2 and Q mentioned in Lemma 6.2.
(2) 1) and ( 2) imply that γ(Q 1 ) = γ(P ′ 1 ) and, similarly, the equations ( 2) and ( 3) imply that γ(Q 3 ) = γ(P ′ 2 ).But these imply that 0 = γ(P we have that Q i ⊆ P i , and the paths in i∈[t] Q i are vertex-disjoint, as required.Lemma 6.1 mentions a set of vertex-disjoint V =2 (W )-paths in G, but note that these may arbitrarily intersect the internal vertices of corridors of W .The following technical lemma allows us to take subpaths of these paths which intersect the corridors of W in a more controlled manner.Lemma 6.3.Let Γ be an abelian group, let (G, γ) be a Γ-labelled graph and let H ⊆ G be a subdivision of a 3-connected graph such that every corridor of H is γ-zero.If G contains a γ-nonzero V =2 (H)-path P , then there exist a subpath U of P and a set X of at most 12 corridors of H satisfying the following properties: (i) H ∩ U is a subgraph of X .
(ii) For any subgraph H ′ ⊆ H which is a subdivision of a 3-connected graph with X ⊆ H ′ , and any subset Proof.For each vertex z of H, we define x z,1 , x z,2 ∈ Γ and a path X z as follows.
• If z has degree 2 in H, then let X z be the corridor of H containing z, let x z,1 := γ(X z,1 ), and x z,2 := γ(X z,2 ) where X z,1 and X z,2 are the two distinct subpaths of X z from z to the endvertices of X z .• Otherwise let X z be a path of length 0 containing z, let x z,1 := 0, and and is γ-breaking otherwise.We first prove the following claim.
Claim.P ∪ H contains a γ-breaking path U such that (a) both endvertices of U are in H, (b) at most two corridors of H intersect the set of internal vertices of U , and (c) for each endvertex z of U , either z ∈ V =2 (H) or X z contains no internal vertex of U .
Proof.Suppose that this claim does not hold.We first show that ( * ) if P contains a V (H)-path Q from a to b where X a = X b , then Q is a γ-preserving path.Because there are no two distinct corridors of H with the same set of endvertices, X a intersects at most one of X b,1 and X b,2 .If γ(X a,1 ) = γ(X a,2 ) and X b,i does not intersect X a for some i ∈ {1, 2}, then X a,1 ∪ Q ∪ X b,i or X a,2 ∪ Q ∪ X b,i is γ-breaking.It is not difficult to verify that such a γ-breaking path satisfies the required properties, contradicting the assumption.Thus, γ(X a,1 ) = γ(X a,2 ), and by symmetry, γ(X b,1 ) = γ(X b,2 ).Now, by the assumption, Q is γ-preserving.This shows ( * ).
Let M 1 be the set of V (H)-paths in P whose endvertices are internal vertices of distinct corridors of H. Let M 2 be the set of maximal subpaths of P − Q∈M 1 E(Q) of length at least 1.Note that for each R ∈ M 2 , at most one corridor of H intersects the set of internal vertices of R.
Let v and w be the endvertices of P , and let t : a partition of P into t edge-disjoint subpaths, each having length at least 1.Let P 1 be the path in M 1 ∪ M 2 containing v, and for each i ∈ [t − 1] let P i+1 be the unique path in (M 1 ∪ M 2 ) \ {P j : j ∈ [i]} sharing an endvertex, say v i , with P i .By ( * ), we have γ( We claim that there is a γ-breaking path in M 2 .Suppose for a contradiction that all paths in M 2 are γ-preserving.By ( * ), all paths in M 1 are γ-preserving and therefore As every corridor of H is γ-zero, we know that This implies that t−1 i=0 γ(P i+1 ) = γ(P ) = 0, which contradicts the fact that P is γ-non-zero.So, we conclude that there exists j ∈ [t] such that P j ∈ M 2 and P j is γ-breaking.
We obtain that the path P ′ defined by has the desired properties.
Let U be a path obtained by the previous claim.Let X 1 be the set of corridors of H intersecting the set of internal vertices of U .By the previous claim, we have Let a, b be the endvertices of U .For x ∈ {a, b}, let Y x be a corridor of H containing x. Thus if x has degree 2 in H, then Y x = X x and otherwise Y x is an arbitrary corridor of H ending at x. Let X 2 be a minimal set of corridors of H such that Y a , Y b ∈ X 2 and each endvertex of Y a and Y b is contained in at least three corridors in X 2 .Then |X 2 | ≤ 10.
Let X := X 1 ∪ X 2 .Then |X | ≤ 12 and (i) holds.It remains to show (ii).Let H ′ be a subgraph of H which is a subdivision of a 3-connected graph Ĥ′ such that X is a subgraph of H ′ and let T be subset of V =2 (H ′ ) of size at least 3.Note that V =2 (H ′ ) ⊆ V =2 (H) and therefore every corridor of H ′ is γ-zero.By the construction of X 2 , each endvertex of U is contained in some corridor of H which is also a corridor of H ′ and every corridor of H intersecting the set of internal vertices of U is also a corridor of H ′ .Hence from the claim, we deduce that (a ′ ) both endvertices of U are in H ′ , (b ′ ) at most two corridors of H ′ intersect the set of internal vertices of U , and (c ′ ) for each endvertex z of U , either z ∈ V =2 (H ′ ) or the corridor of H ′ containing z contains no internal vertex of U .Since Ĥ′ is 3-connected, there are two disjoint paths Then x is not an endvertex of U .Since x is in at least 3 corridors of H ′ , by property (b ′ ), x / ∈ V (U ).Since Ĥ′ is 3-connected, by properties (b ′ ) and (c ′ ), H ′ − E( X 1 ) is connected.Thus, H ′ has a path Q connecting the endvertices of U which contains no internal vertex of U .The cycle O := Q ∪ U is γ-non-zero since U is γ-breaking.Since Ĥ′ is 3-connected, there are three vertex-disjoint paths from T to {x} ∪ V =2 (U ) in H ′ .By extending one of the paths ending at x to an internal vertex of U through R if x / ∈ V (O), we obtain three vertex-disjoint (V (O), T )-paths in H ′ ∪ U .Hence Lemma 4.2 yields the desired result.
In the next lemma, we extend subpaths from the previous lemma to handles of some suitable column-slice.Lemma 6.4.There exist functions w 6.4 : N 2 → N and f 6.4 : N → N satisfying the following.Let k and c be positive integers with c ≥ 3, let Γ be an abelian group, and let (G, γ) be a Γ-labelled graph.Let W be a wall in G of order at least w 6.4 (k, c) such that all corridors of W are γ-zero.
Claim.For all i ∈ [h(k)] there exist a set C i of 3-column-slices of W , a set R i of 3-row-slices of W and a subpath U i of a path in P such that, with U i and U j are vertex-disjoint for all j ∈ [i − 1], (e) every 3-column-slice of W that intersects U i also intersects some 3-column-slice in j∈[i] C j , (f ) for any column-slice W ′ of W which is disjoint from U i , there is a γ-non-zero W ′ -handle in Proof of Claim.We proceed by induction on i.For i ∈ [h(k)], assume that the claim holds for all j ∈ [i − 1].We define • C i to be the set of all 3-column-slices of W which intersect no 3-column-slices in j∈[i−1] C j , • R i to be the set of all 3-row-slices of W which intersect no 3-row-slices in j∈[i−1] R j , and We will first show that the number of vertices in V =2 (W ) \ V ( H i ) is small.Let I be the set of all column indices a such that the a-th column C W a intersects no 3-column-slice in j∈[i−1] C j .Then I admits a partition into intervals consisting of consecutive integers such that the number of intervals is bounded by j∈ Observe that at least one interval of I has size at least 3 because w x intersects some 3-column-slice in C j for some j < i or x belongs to an interval of I of size at most 2. Since at least one interval of I has size more than 2, the number of possible values of x is at most 3 By the same argument, we deduce that R i is nonempty and the number of possible values of y is at most 240(i − 1).Thus, the number of vertices in V =2 (W ) not in H i is at most 2(240(i − 1)) 2 , because there are at most two vertices of , there is a path P i in P both of whose endvertices are in H i such that U j is not a subpath of P i for all j < i.Let X i be the set of at most 12 corridors of H i and let U i be a subpath of P i guaranteed by Lemma 6.3.
Let C i be a minimal non-empty subset of C i containing all 3-column-slices in C i which intersect some corridor in X i .Since each corridor of H i intersects at most four 3-column-slices, Similarly, let R i be a minimal non-empty subset of R i containing all 3-row-slices in R i which intersect some corridor in X i .Then |R i | ≤ 48.Now (a)-(e) are true by construction.
To see (f), let W ′ be a column-slice disjoint from U i .Note that H i ∪ W ′ is a subdivision of a 3-connected graph.Applying Lemma 6.3(ii) with Let I be the set of all column indices a such that C W a intersects no 3-column-slice in j∈[h(k)] C j .By (a), I admits a partition into at most 48h(k) + 1 disjoint intervals, each consisting of consecutive integers.Since w any vertex set of size at most f 2.9 (k) cannot hit all these paths.By Theorem 2.9, there exist k vertex-disjoint γ-non-zero W ′ -handles, as desired.Now we obtain Lemma 6.1 as a corollary of Lemmas 6.2 and 6.4.
Proof of Lemma 6.1.Let w 6.1 and f 6.1 be the functions w 6.4 and f 6.4 respectively, as given by Lemma 6.4.By Lemma 6.4, there exists a c-column-slice W ′ of W and a set P ′ of k vertex-disjoint γ-non-zero W ′ -handles.For each i ∈ [t − 1] and each P ∈ P i , let R P,1 and R P,2 be the rows of W containing the endvertices of P , and let Q P be the row-extension of P to W ′ , that is the unique V (W ′ )-path such that P ⊆ Q P ⊆ P ∪ R P,1 ∪ R P,2 .We remark that it is possible that R P,1 = R P,2 .Now applying Lemma 6.2 to the sets {Q P : P ∈ P i } for i ∈ [t − 1] together with P ′ yields the desired result.

Basic lemmas for products of abelian groups
In this section, we prove some basic lemmas on products of abelian groups that will be useful throughout Sections 8 and 9.
An arithmetic progression is a set of integers A such that there are integers a and b = 0 for which A = {a + bn : n ∈ Z}.For a set A = {A i : i ∈ [k]} of arithmetic progressions, we say A covers a set S if S ⊆ i∈[k] A i .We will use the following fact about arithmetic progressions, conjectured by Erdős in 1962 and proven by Crittenden and Vanden Eynden [3] in 1969.We cite an equivalent version of Balister et al. [1], who presented a simple proof.
Corollary 7.2.Let m, t, and ω be positive integers, let Γ = j∈[m] Γ j be a product of m abelian groups, and for all j ∈ [m] let Ω j be a subset of Γ j of size at most ω.For all i ∈ [t] and j ∈ [m], let g i,j be an element of Γ j .If for each i ∈ [t] there exists an integer c i such that t i=1 c i g i,j / ∈ Ω j for all j ∈ [m], then for each i ∈ [t] there exists an integer Proof.Pick an integer d i for each i ∈ [t] such that t i=1 d i g i,j / ∈ Ω j for all j ∈ [m], and subject to this |{i ∈ [t] : d i ∈ [2 mω ]}| is maximised.Suppose for a contradiction that for some x ∈ [t], we have that d x / ∈ [2 mω ].Without loss of generality, we may assume x = t.For all j ∈ [m] and g ∈ Ω j , let A j,g be the set of integers d such that dg t,j + t−1 i=1 d i g i,j = g.Note that A j,g is an arithmetic progression or contains at most one integer.Let A ′ j,g be an arithmetic progression such that A j,g ⊆ A ′ j,g and d t / ∈ A ′ j,g .Such an A ′ j,g exists; if A j,g is an arithmetic progression then let A ′ j,g := A j,g , and if A j,g contains a unique integer a j , then let A ′ j,g be the arithmetic progression {a j + 2(d t − a j )k : k ∈ Z}.For a sequence a = (a i : i ∈ [t]) over an abelian group Γ, we let Σ(a) denote the set of all sums of subsequences of a.We write |a| := t, the length of a.We say a ∈ Γ is repeated in a if a = a i = a j for some 1 ≤ i < j ≤ t.We say that a is good if |Σ(a)| ≥ |a|.Obviously, a sequence of pairwise distinct elements of Γ is good.Also, observe that for an element g of order at least t, if a i := g for all i ∈ [t], then the sequence a = (a i : Lemma 7.3.Let Γ be an abelian group and let a = (a i : i ∈ [t]) be a sequence of length t over Γ.If all repeated elements in a have order at least t, then a is good.
Proof.We proceed by induction on t.We may assume that t ≥ 2. If a has no repeated elements, then {a i : i ∈ [t]} ⊆ Σ(a) and therefore a is good.Thus, without loss of generality, we may assume that a t is a repeated element.Let a ′ := (a i : i ∈ [t − 1]) and S := Σ(a ′ ).By induction |S| ≥ t − 1.We may assume that |S| = t − 1.Let T := {x + a t : x ∈ S} ⊆ Σ(a).If S = T , then x∈S x = x∈S (x + a t ) and therefore |S|a t = 0, contradicting the assumption on the order of a t .Thus S = T and therefore |Σ(a)| ≥ |S ∪ T | ≥ t.
The following lemma and its corollary are useful to find a cycle whose γ i -value is not in Ω i for all i ∈ [m].Lemma 7.4.Let m, t, and ω be positive integers, let Γ = j∈[m] Γ j be a product of m abelian groups and for all j ∈ [m] let Ω j be a subset of Γ j of size at most ω.For all i ∈ [t] let S i be a subset of Γ such that for each j ∈ [m] there exists some i ∈ [t] such that π j (g) = π j (g ′ ) for all distinct g, g ′ in S i .If |S i | > mω for all i ∈ [t], then for every h ∈ Γ there is a sequence , and (ii) Proof.Uniformly at random, select g i ∈ S i independently for each i ∈ [t], and consider the sum g := h + i∈[t] g i .For each j ∈ [m], there exists i ∈ [t] such that g and every group element obtained by replacing g i in the sum with a different element of S i have distinct j-th coordinates.Hence, the probability that π j (g) ∈ Ω j is at most ω/(mω + 1).It follows that there is a positive probability that π j (g) / ∈ Ω j for all j ∈ [m].
For two sequences a = (a i : i ) of length t over a product Γ = j∈[m] Γ j of m abelian groups, we write a − b to denote the sequence (a i − b i : i ∈ [t]) and for j ∈ [m], we write π j (a) to denote the sequence (π j (a i ) : i ∈ [t]) over Γ j .Corollary 7.5.Let m and ω be positive integers, let Γ = i∈[m] Γ i be a product of m abelian groups, and for each i ∈ [m] let Ω i be a subset of Γ i of size at most ω and let a i := (a i,j : j ∈ [mω + 1]) and b i := (b i,j : j ∈ [mω + 1]) be two sequences over Γ such that π i (a i − b i ) is good.Then for all h ∈ Γ, i ∈ [m], and j ∈ [mω + 1], there exists c i,j ∈ {a i,j , b i,j } such that for all x ∈ [m], we have π Proof.By definition of a good sequence, for each i ∈ [m], we have . We apply Lemma 7.4 with h ′ to find a sequence (g i : i ∈ [m]) such that g i ∈ S i for each i ∈ [m] and we have that g i ∈ Σ(a i − b i ), and hence for each j ∈ [mω + 1] there exists c i,j ∈ {a i,j , b i,j } such that g i + j∈[mω+1] b i,j = j∈[mω+1] c i,j .This completes the proof.
Given positive integers n and k, we write R(n; k) for the minimum integer N such that in every kcolouring of the edges of K N there is a monochromatic copy of K n .A classical result of Ramsey [15] shows that R(n; k) exists.
Lemma 7.6.There exists a function f 7.6 : N 2 → N satisfying the following.Let m, t and N be positive integers with N ≥ f 7.6 (t, m) and let Γ = i∈[m] Γ i be a product of m abelian groups.Then for every sequence (g i : i ∈ [N ]) over Γ, there exists a subset such that for all distinct i and j in [N ] there exists x ∈ Z such that π x (g i ) = π x (g j ), then the second condition holds for some i ∈ Z.
Proof.Let f 7.6 (t, m) := R(t; 2 m ).We define a 2 m -colouring of the edges of K N by colouring each edge xy of K N by the set {i ∈ [m] : π i (g x ) = π i (g y )}.The result follows from the definition of R(t; 2 m ).Note that if Z is subset of [m] such that for all distinct i and j in [N ] there exists an x ∈ Z such that π x (g i ) = π x (g j ), then every set used in the colouring intersects Z.

From handles to cycles
The focus of this section is proving the following key lemma, which will be the final ingredient needed in the proof of Theorem 1.1 for constructing the cycles from the clean subwall from Section 5 and the sets of handles from Section 6. Lemma 8.1.There exist functions c 8.1 , r 8.1 : N 4 → N satisfying the following.Let t, ℓ, m and ω be positive integers with ℓ ≥ 3, let Γ = i∈[m] Γ i be a product of m abelian groups, for each i ∈ [m] let Ω i be a subset of Γ i of size at most ω, and let (G, γ) be a Γ-labelled graph.Let Z be a subset of [m], let W be a (γ, Z, ℓ)-clean (c, r)-wall with c ≥ c 8.1 (t, ℓ, m, ω) and r ≥ r 8.1 (t, ℓ, m, ω).For every set P of at most t vertex-disjoint W -handles such that γ i ( P) / We begin by linking up a set of handles of a wall into a cycle.
Lemma 8.2.Let t be a positive integer and let W be a (c, r)-wall in a graph G with r ≥ 3 and c ≥ max{3, t + 1}.For every set P of at most t vertex-disjoint W -handles in G, there is a cycle O in W ∪ P that contains P as a subgraph.
Proof.Let T be the set of endvertices of all paths in P. We proceed by induction on t.If |P| ≤ 2, then as W ∪ P is 2-connected, W ∪ P has a cycle O containing at least one edge from every path in P. Therefore, we may assume that |P| = t > 2.
By symmetry, we may assume that the first column of W meets at least two paths in P. In the first column of W , choose two degree-2 nails v 1 , v 2 that are endvertices of distinct paths P 1 , P 2 of P respectively such that the distance between v 1 and v 2 in the first column of W is minimised.Let Q be the path from v 1 to v 2 in the first column of W .Let P * be the path P 1 ∪ Q ∪ P 2 .Let W ′ be the (c − 1)-column-slice of W obtained by removing the first column.Let P ′ be the row-extension of (P \ {P 1 , P 2 }) ∪ {P * } to W ′ .Since P ′ is a set of t − 1 vertex-disjoint W ′ -handles, by the induction hypothesis, there is a cycle O in W ′ ∪ P ′ such that P ′ ⊆ O.This completes the proof because P ⊆ P ′ and W ′ ∪ P ′ ⊆ W ∪ P.
In order to obtain a cycle whose γ i -value is not in Ω i for every i ∈ [m], it will be useful to have access to a sequence of subwalls to reroute segments of the cycle constructed by the previous lemma.The following straightforward corollary provides this.The case P = ∅ is easy to verify, so we may assume |P| = t > 0. Without loss of generality, we may assume that the last column of W intersects P. Let W ′′ be a kw-column-slice of W containing the last column of W and let W ′ be a (c − kw)-column-slice of W disjoint with W ′′ .Let P ′ denote the row-extension of P to W ′ in W .
By the pigeonhole principle, there is a (w − 1)-row-slice of W ′′ which is disjoint from P ′ .Hence there is a w-row-slice S of W ′′ such that S ∩ P ′ = R S 1 .We can pack k vertex-disjoint N W -anchored (w, w)-subwalls Applying Lemma 8.2 to W ′ and P ′ yields the desired cycle.Moreover, we need the following variation of Lemma 4.2.
Lemma 8.4.Let Γ be an abelian group, and let (G, γ) be a Γ-labelled graph, let O be a γ-non-zero cycle in G, and let P be a path disjoint from O. If G contains three vertex-disjoint (V (P ), V (O))paths P 1 , P 2 , P 3 , then there is a path P ′ in P ∪ O ∪ P 1 ∪ P 2 ∪ P 3 with the same endvertices as P such that γ(P ′ ) = γ(P ).
Finally, we can prove Lemma 8.1.
For each i ∈ [y], we now recursively define a family Q i,j : j ∈ [yω + 1] of paths and a family (S i,j : j ∈ [yω + 1]) of subsets of Γ i , such that for all j ∈ [yω + 1] and g ∈ S i,j , • |S i,j | ≤ j − 1, and • the order of g is at most mω.We first set S i,1 := ∅.Now, for j ∈ [yω + 1], let λ j be the induced Γ i / S i,j -labelling of G.Note that since i / ∈ Z and W is (γ, Z, ℓ)-clean, W has no γ i -bipartite (ℓ, ℓ)-subwall, and in particular, each W ′ i,j does not have such a subwall.As |S i,j | ≤ mω and each element of S i,j has order at most mω, Corollary 5.4 implies that there is a λ j -non-zero cycle O i,j in W ′ i,j .By Lemma 8.4, there is a path Q i,j in W i,j with the same endvertices as P i,j such that λ j (P i,j ) = λ j (Q i,j ), since there are three vertex-disjoint (V (P i,j ), V (O i,j ))-paths in W i,j .We set ) has order at least mω + 1, and . By construction of the sets S i,j , the projection of D i to Γ i contains no repeated elements of order at most mω in Γ i , and thus this sequence is good by Lemma 7.3.Hence, by Corollary 7.5 with h := γ(H), there exists X i,j ∈ {P i,j , Q i,j } for all i ∈ [y] and j ∈ [yω + 1] such that for the cycle O

Proof of the main theorem
We now complete the proof of Theorem 1.1, which we are restating for the convenience of the reader.
Theorem 1.1.For every pair of positive integers m and ω, there is a function f m,ω : N → N satisfying the following property.For each i ∈ [m], let Γ i be an abelian group, and let Ω i be a subset of Γ i .Let G be a graph and for each i ∈ [m], let γ i be a Γ i -labelling of G, and let O be the set of all cycles of G whose γ i -value is in , then for all k ∈ N there exists either a half-integral packing of k cycles in O, or a hitting set for O of size at most f m,ω (k).
Proof.Let Γ := i∈[m] Γ i denote the product of the m given abelian groups, and let γ : E(G) → Γ denote the Γ-labelling for which γ i (e) = π i (γ(e)) for all e ∈ E(G).We proceed by induction on k.For k ≤ 2, we may trivially set f m,ω (k) = 1.Now suppose that k > 2 and that there is some integer f m,ω (k − 1) as per the theorem.For every subgraph H of G, let ν(H) denote the maximum size of a set of cycles O in H with γ i (O) / ∈ Ω i for all i ∈ [m] such that no three cycles in the set share a common vertex.Observe that ν is a packing function for G.We will show that if τ ν (G) is sufficiently large relative to k, then ν(G) ≥ k.This will complete the proof of the theorem.
We now show that W ′′ , (P ′′ ) is either a (c ′ , q, p + 1)-McGuffin or a (c ′ , q + 1, q + 1)-McGuffin, contradicting the maximality of (q, p).First, observe that since W is (γ ′ , Z, ψ(|Z| + 1) + 2)-clean, every N W -path is γ i -zero for all i ∈ Z and therefore if P ′ is the row-extension of P to W ′′ in W ′ , then γ i (P ′ ) = γ i (P ) for all i ∈ Z, implying (d) and (e) for i < p ′′ .By the definition of Z ′ , properties (d) and (e) hold for i = p ′′ .It remains to check (h) when Z ′ is empty, q < i ≤ p, and i ′ = p ′′ = p + 1.This is implied by the property that the paths in P ′ p+1 are λ-non-zero.and for j ∈ I define S j := {γ(P ) : P ∈ P j \ (X i,j ∪ Y j )}.By Claim 1(d), for all j ∈ I, for all distinct g, g ′ in S j , and for all x ∈ Z j , we have π x (g) = π x (g ′ ) and so |S j | > mw.Now by Lemma 7.4, there is a family (Q j : j ∈ I) of paths such that Q j ∈ P j \ (X i,j ∪ Y j ) for all j ∈ I, and π x h + j∈I γ(Q j ) / ∈ Ω x for all x ∈ Z.Hence, let Q i := {Q j : j ∈ I} ∪ j∈[p] Y j , and note that |Q i | ≤ j∈[p] d j ≤ 2 |Z 0 |w p.
We now complete the proof of the theorem.Since W ′ has at least β(1, 0, |Z|) columns, there is a set {W i : i ∈ [k]} of k vertex-disjoint c 8.1 (2 mω p, ψ(|Z| + 1) + 2, m, ω)-column-slices of W ′ .Note that the number of rows of W ′ is at least ψ(|Z|) ≥ r 8.1 (2 mω p, ψ(|Z| + 1) + 2, m, ω).For each i ∈ [k], let Q * i be the row-extension of Q i to W i .By Lemma 8.1, for each i ∈ [k] there is a cycle O i in W i ∪ Q * i with γ j (O i ) / ∈ Ω j for all j ∈ [m].Observe that for every vertex v ∈ V (G), there are at most two indices i ∈ [k] such that v ∈ V (W i ∪ Q * i ).Hence, ν(G) ≥ k, a contradiction.

Conclusion
In this work, we proved that a half-integral analogue of the Erdős-Pósa theorem holds for cycles in graphs labelled with a bounded number of abelian groups, whose values avoid a bounded number of elements of each group.We conclude with some open problems.
In the proof of our theorem, the theorem of Wollan [28] about γ-non-zero A-paths was important.This theorem implies that an analogue of the Erdős-Pósa theorem holds for the odd A-paths, and for A-paths intersecting a prescribed set of vertices.Bruhn, Heinlein, and Joos [2] further showed that an analogue of the Erdős-Pósa theorem holds for A-paths of length at least ℓ, and for Apaths of even length, but interestingly, they also showed that for every composite integer m > 4 and every d ∈ {0} ∪ [m − 1], no such analogue holds for A-paths of length d modulo m.Later, Thomas and Yoo [24] characterised the abelian groups Γ and elements ℓ ∈ Γ where an analogue of the Erdős-Pósa theorem holds for A-paths of γ-value ℓ.We would like to ask whether a statement similar to Theorem 1.1 holds for A-paths.Question 1.For every pair of positive integers m and ω, does there exist a function g m,ω : N → N satisfying the following property?
• For each i ∈ [m], let Γ i be an abelian group and let Ω i be a subset of Γ i .Let G be a graph, let A be a set of vertices in G, and for each i ∈ [m], let γ i be a Γ i -labelling of G, and let P be the set of all A-paths of G whose γ i -value is in Γ i \ Ω i for all i ∈ [m].If |Ω i | ≤ ω for all i ∈ [m], then there exists either a half-integral packing of k paths in P, or a hitting set for P of size at most g m,ω (k).
Our next question relates to directed labellings of graphs.Let Γ be a group (not necessarily abelian).A directed Γ-labelling of a graph G is a function γ from the set of oriented edges E(G) := {(e, w) : e = uv ∈ E(G), w ∈ {u, v}} to Γ such that γ(e, u) = −γ(e, v) for each edge e = uv.Given a walk W := v t e 1 v 1 e 2 • • • e t v t , we define γ(W ) := t j=1 γ(e j , v j ), and say that W corresponds to a cycle O if E(O) = E(W ) and v t is the only repeated vertex of W .It is straightforward to check that if any walk corresponding to a cycle O has value 0, then all walks corresponding to O do, so it makes sense to consider non-zero cycles with respect to a directed labelling.Note that if Γ is abelian and W 1 and W 2 are walks corresponding to the same cycle O, then γ(O 1 ) = ±γ(O 2 ).If Γ is not abelian, then the choice of start vertex for the corresponding walk does matter as well.Hence, in this case, we are really considering cycles together with a specified start vertex and direction.
Huynh, Joos, and Wollan [10] conjectured that a half-integral analogue of the Erdős-Pósa theorem holds for cycles which are non-zero with respect to a fixed number of directed labellings.We ask whether a statement similar to Theorem 1.1 holds for directed labellings.If it does, then it would imply the conjecture of Huynh, Joos, and Wollan.As discussed in Section 3.2, an analogue of the Erdős-Pósa theorem does not hold for the cycles described in Theorem 1.1.It was shown that in graphs of sufficiently high connectivity [27,16,13,11], an analogue of the Erdős-Pósa theorem holds for odd cycles.We ask whether a similar phenomenon happens for the cycles described in Theorem 1.1.Question 3.For every pair of positive integers m and ω, do there exist functions g m,ω : N → N and g ′ m,ω : N → N satisfying the following property?
• For each i ∈ [m], let Γ i be an abelian group and let Ω i be a subset of Γ i .Let G be a graph, and for each i ∈ [m], let γ i be a Γ i -labelling of G, and let O be the set of all cycles of G whose γ i -value is in Similar to Question 3, we ask whether an analogue of the Erdős-Pósa theorem holds for the A-paths described in Question 1, and for the cycles described in Question 2, in the case of highly connected graphs.

Corollary 1 . 5 .
Let Σ be a compact surface with Z 2 -homology group Γ and let C be a set of Z 2homology classes of Σ.Let G be a graph embedded on Σ, and let O be the set of all cycles of G whose Z 2 -homology classes are contained in C. Then for all k ∈ N there exists either a half-integral packing of k cycles in O, or a hitting set for O of size at most f 1,|Γ|−|C| (k).
For a family F = (x i : i ∈ I) we write |F| = |I|, called the size of F. 2.1.Walls.Let c, r ≥ 3 be integers.The elementary (c, r)-wall W c,r is the graph obtained from the graph on the vertex set [2c] × [r] whose edge set is

Figure 2 .
Figure 2. A 3-row-slice W 3 , a 4-row-slice W 4 , and a 5-row-slice W 5 of a (5, 6)wall W . Notice that the first column of W 3 is in the first column of W but the first column of W 4 or W 5 is in the last column of W .

Corollary 3 . 3 .
For any function f m,ω as in Theorem 1.1, we have f m,ω (3) > (2 + mω)/2.Proof.Consider for each i ∈ [m] the group Γ i := Z and the subset Ω and so by the minimality of T and the fact that|S ∪ Y | ≤ |S| + |Y | < |X| + |Z| + |Y | = |T | we have that ν(G − B) ≥ 1.By symmetry ν(G − A) ≥ 1, and since ν(G − A) + ν(G − B) ≤ ν(G), both ν(G − A) and ν(G − B) are strictly less than ν(G).By the assumption, τ ν (G − A), τ ν (G − B) ≤ t/12.Let T A and T B be ν-hitting sets of minimum size for G − A and G − B, respectively.Then contradicting the assumption that T is a minimum ν-hitting set.Let (A, B) be a separation of order less than t/6 with |B ∩ T | ≥ |A ∩ T |, and let S := A ∩ B. Clearly, (B, A) / ∈ T T .Suppose for a contradiction that (A, B) / ∈ T T and hence A \ B contains at least t/6 vertices of T .Since B \ A contains at least as many vertices of T as A \ B does, by the claim there is a linkage of size ⌈t/6⌉ in G from A \ B to B \ A, contradicting the assumption on the order of (A, B).Hence, (A, B) ∈ T T .Note that |T ∩ A| < t/3 for each (A, B) ∈ T T .Hence for (A 1 , B 2 ), (A 2 , B 2 ), (A 3 , B 3 )∈ T T we have that |T ∩ (A 1 ∪ A 2 ∪ A 3 )| < t, and hence G[A 1 ] ∪ G[A 2 ] ∪ G[A 3 ] = G.Thus we conclude that T T is a tangle of order ⌈t/6⌉.
of size less than u − 2 hitting all γ-non-zero N -paths.Since |S| < u ≤ τ ν (G), we have that ν(G − S) ≥ 1, so G−S has a γ-non-zero cycle O with ν(O) ≥ 1.By Lemma 4.2, G−S does not have three vertex-disjoint (V (O), N )-paths.By Menger's theorem applied to G − S, there exists S ′ ⊆ V (G) of size at most |S| + 2 separating O from N .Since |S ′ | < u, by the given assumption on N , the graph G − S ′ has a component H containing a vertex of N and at least 4u vertices of T

Claim 2 .
There is a family (Q i : i ∈ [k]) of k disjoint subsets of i∈[p] P i , each of size at most 2 |Z 0 |ω p, such that for each i ∈ [k] and each j ∈ Z, we haveγ j Q i / ∈ Ω j .Proof.Recursively, for each i ∈ [k] we define Q i containing at most 2 |Z 0 |ω elements of P j for all j ∈ [p].For each i ∈ [k] and j ∈ [p], let X i,j := P j ∩ i ′ ∈[i−1] Q i ′ , and note that we have X 1,j = ∅ and |X i,j | ≤ (i − 1)2 |Z 0 |w ≤ (k − 1)2 mw .For each j ∈ [p], select g j ∈ γ(P j ) arbitrarily.By Claim 1(e) and (f), for each j ∈ [p] there exists an integer c j such that π x j∈[p] c j g j / ∈ Ω x for all x ∈ Z 0 .Hence, by Corollary 7.2, for each j ∈ [p]there exists an integer d j ∈ 2 |Z 0 |ω such that for all x ∈ Z 0 , we haveπ x j∈[p] d j g j / ∈ Ω x .Let I be the set of indices j ∈ [p] such that Z j = ∅.Now |P j | ≥ α(p, |Z 0 |) ≥ α(p, 0) ≥ k2 mw ≥ |X i,j | + d j for all j ∈ [p]by Claim 1(c).Hence, for each j ∈ [p], we can select a set Y j of distinct W ′handles in P j \ X i,j of size d j − 1 if j ∈ I and of size d j otherwise.By the definition of X i,j and Claim 1(c), there are at least α(p, |Z 0 |) − k2 |Z 0 |ω ≥ mω + 1 distinct W ′ -handles in P j \ (X i ∪ Y j ) for every j ∈ [p].Define h := j∈[p] P ∈Y j γ(P ),

Question 2 .
For every pair of positive integers m and ω, does there exist a function g m,ω : N → N satisfying the following property?•For each i ∈ [m], let Γ i be a group and let Ω i be a subset of Γ i .Let G be a graph and for each i ∈ [m], let γ i be a directed Γ i -labelling of G, and let O be the set of all cycles of G which have a corresponding walk W such that γ i (W ) is inΓ i \ Ω i for all i ∈ [m].If |Ω i | ≤ ω for all i ∈ [m],then there exists either a half-integral packing of k cycles in O, or a hitting set for O of size at most g m,ω (k).
k)-connected and |Ω i | ≤ ω for all i ∈ [m],then there exists either a set of k vertex-disjoint cycles in O, or a hitting set for O of size at most g m,ω (k).
is a set of mω arithmetic progressions not covering d t .By Theorem 7.1, there exists d ′ t ∈ [2 mω ] such that d ′ t g t,j + t−1 i=1 d i g i,j / ∈ Ω j for all j ∈ [m], contradicting our choice of d t .
Corollary 8.3.There exist functions c 8.3 , r 8.3 : N 3 → N satisfying the following.Let t, k and w be positive integers with w ≥ 4. Let G be a graph containing a (c, r)-wall W with c ≥ c 8.3 (t, k, w) and r ≥ r 8.3 (t, k, w).For every set P of at most t vertex-disjoint W -handles in G, there exist a cycle O in W ∪ P and a set {W i