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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M. Chudnovsky, J. Geelen, B. Gerards, L. Goddyn, M. Lohman and P. Seymour: Packing non-zero A-paths in group labeled graphs, Combinatorica 26(5) (2006), 521–532.
I. Dejter and V. Neumann-Lara: Unboundedness for generalized odd cycle traversability and a Gallai conjecture, paper presented at the Fourth Caribbean Conference on Computing, Puerto Rico, 1985.
P. Erdős and G. Szekeres: A combinatorial problem in geometry, Compositio Mathematica 2 (1935), 463–470.
T. Gallai: Maximum-minimum Sätze and verallgemeinerte Factoren von Graphen, Acta. Math. Hung. Acad. Sci. 12 (1961), 131–173.
M. Kriesell: Disjoint A-paths in digraphs, J. Combin. Theory, Ser. B 95 (2005), 168–172.
P. Erdős and L. Pósa: On the maximal number of disjoint circuits of a graph, Publ. Math. Debrecen 9 (1962), 3–12.
K. Kawarabayashi and A. Nakamoto: The Erdős-Pósa property for vertex- and edge-disjoint odd cycles on an orientable fixed surface, Discrete Math. 307(6) (2007), 764–768.
K. Kawarabayashi and B. Reed: Highly parity linked graphs, Combinatorica 29(2) (2009), 215–225.
K. Kawarabayashi and P. Wollan: Non-zero disjoint cycles in highly connected group labeled graphs, J. Combin. Theory, Ser. B 96 (2006), 754–757.
W. Mader: Über die Maximalzahl krezungsfreier H-Wege, Archiv der Mathematik (Basel) 31 (1978), 387–402.
R. Rado: Covering theorems for ordered sets, Proceedings of the London Math. Soc. s2-50(7) (1949), 509–535.
D. Rautenbach and B. Reed: The Erdős-Pósa property for odd cycles in highly connected graphs, Combinatorica 21(2) (2001), 267–278.
B. Reed: Mangoes and blueberries, Combinatorica 19(2) (1999), 267–296.
N. Robertson and P. Seymour: Graph Minors X, Obstructions to a tree-decomposition; J. Combin. Theory, Ser. B 52 (1991), 153–190.
N. Robertson, P. Seymour and R. Thomas: Quickly excluding a planar graph, J. Combin. Thoery, Ser. B 62 (1994), 323–348.
C. Thomassen: On the presence of disjoint subgraphs of a specified type, J. Graph Theory 12 (1988), 101–110.
C. Thomassen: The Erdős-Pósa property for odd cycles in graphs with large connectivity, Combinatorica 21(2) (2001), 321–333.
P. Wollan: Packing non-zero A-paths in an undirected model of group labeled graphs, J. Combin. Thoery, Ser. B 100 (2010), 141–150.
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This work partially supported by a fellowship from the Alexander von Humboldt Foundation.