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

Authors Ken-ichi Kawarabayashi, Yusuke Kobayashi

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Ken-ichi Kawarabayashi
Yusuke Kobayashi

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Ken-ichi Kawarabayashi and Yusuke Kobayashi. Edge-disjoint Odd Cycles in 4-edge-connected Graphs. In 29th International Symposium on Theoretical Aspects of Computer Science (STACS 2012). Leibniz International Proceedings in Informatics (LIPIcs), Volume 14, pp. 206-217, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2012)


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.
  • odd-cycles
  • disjoint paths problem
  • Erdös-Posa property
  • packing algorithm
  • 4-edge-connectivity


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