## Abstract

We prove that for all positive integers *k*, there exists an integer *N* =*N*(*k*) such that the following holds. Let *G* be a graph and let *Γ* an abelian group with no element of order two. Let *γ*: *E*(*G*)→*Γ* be a function from the edges of *G* to the elements of *Γ*. A *non-zero* cycle is a cycle *C* such that Σ_{
e∈E(C)}
*γ*(*e*) ≠ 0 where 0 is the identity element of *Γ*. Then *G* either contains *k* vertex disjoint non-zero cycles or there exists a set *X* ⊆ *V* (*G*) with |*X*| ≤*N*(*k*) such that *G−X* contains no non-zero cycle.

An immediate consequence is that for all positive odd integers *m*, a graph *G* either contains *k* vertex disjoint cycles of length not congruent to 0 mod *m*, or there exists a set *X* of vertices with |*X*| ≤ *N*(*k*) such that every cycle of *G-X* has length congruent to 0 mod *m*. No such value *N*(*k*) exists when *m* is allowed to be even, as examples due to Reed and Thomassen show.

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This work partially supported by a fellowship from the Alexander von Humboldt Foundation.

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Wollan, P. Packing cycles with modularity constraints.
*Combinatorica* **31, **95 (2011). https://doi.org/10.1007/s00493-011-2551-5

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### Mathematics Subject Classification (2000)

- 05C22
- 05C38