Finding edge-disjoint odd cycles is one of the most important problems#R##N#in graph theory, graph algorithm and combinatorial optimization. In#R##N#fact, it is closely related to the well-known max-cut problem. One of#R##N#the difficulties of this problem is that the Erdos-Posa property does not hold for odd cycles in general. Motivated by this fact, we prove that for any positive integer k, there exists an integer f(k) satisfying the following: For any 4-edge-connected graph G=(V,E),#R##N#either G has edge-disjoint k odd cycles or there exists an edge set F#R##N#subseteq E with |F| <= f(k) such that G-F is bipartite. We note that#R##N#the 4-edge-connectivity is best possible in this statement.#R##N##R##N#Similar approach can be applied to an algorithmic question. Suppose#R##N#that the input graph G is a 4-edge-connected graph  with n vertices.#R##N#We show that, for any epsilon > 0, if k = O ((log log log n)^{1/2-epsilon}), then the edge-disjoint k odd cycle packing in G can be solved in polynomial time of n.
['Ken-ichi Kawarabayashi', 'Yusuke Kobayashi']
Edge-disjoint Odd Cycles in 4-edge-connected Graphs