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 Erdős-Pósa 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 ⊆ 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-edgeconnected graph with n vertices. We show that, for any ε > 0, if k = O((log log logn)1/2−ε), then the edge-disjoint k odd cycle packing in G can be solved in polynomial time of n. 1998 ACM Subject Classification G.2.2 Graph Theory