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

Abstract

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 Erdös-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
subseteq 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.