Packing cycles with modularity constraints

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.