Edge-disjoint Odd Cycles in 4-edge-connected Graphs

Finding edge-disjoint odd cycles is one of the most important problems in graph theory, graph algorithm and combinatorial optimization. In fact, it is closely related to the well-known max-cut problem. One of 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), either G has edge-disjoint k odd cycles or there exists an edge set F subset of E with |F| <= f(k) such that G-F is bipartite. We note that the 4- edge-connectivity is best possible in this statement. Similar approach can be applied to an algorithmic question. Suppose that the input graph G is a 4-edge-connected graph with n vertices. 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.