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

Ken-Ichi Kawarabayashi , Yusuke Kobayashi

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.

Mots clés: odd-cycles; disjoint paths problem; Erdös-Posa property; packing algorithm; 4-edge-connectivity

Domaines: Complexité [cs.CC]; Algorithme et structure de données [cs.DS]

Ken-Ichi Kawarabayashi, Yusuke Kobayashi. Edge-disjoint Odd Cycles in 4-edge-connected Graphs. STACS'12 (29th Symposium on Theoretical Aspects of Computer Science), Feb 2012, Paris, France. pp.206-217. ⟨hal-00678188⟩