{"@context":"https:\/\/schema.org\/","@type":"ScholarlyArticle","@id":"#article7070","name":"Edge-disjoint Odd Cycles in 4-edge-connected Graphs","abstract":"Finding edge-disjoint odd cycles is one of the most important problems\r\nin graph theory, graph algorithm and combinatorial optimization. In\r\nfact, it is closely related to the well-known max-cut problem. One of\r\nthe difficulties of this problem is that the Erd\u00f6s-P\u00f3sa 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),\r\neither G has edge-disjoint k odd cycles or there exists an edge set F\r\nsubseteq E with |F| <= f(k) such that G-F is bipartite. We note that\r\nthe 4-edge-connectivity is best possible in this statement.\r\n\r\nSimilar approach can be applied to an algorithmic question. Suppose\r\nthat the input graph G is a 4-edge-connected graph with n vertices.\r\nWe 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.","keywords":["odd-cycles","disjoint paths problem","Erd\u00f6s-Posa property","packing algorithm","4-edge-connectivity"],"author":[{"@type":"Person","name":"Kawarabayashi, Ken-ichi","givenName":"Ken-ichi","familyName":"Kawarabayashi"},{"@type":"Person","name":"Kobayashi, Yusuke","givenName":"Yusuke","familyName":"Kobayashi"}],"position":20,"pageStart":206,"pageEnd":217,"dateCreated":"2012-02-24","datePublished":"2012-02-24","isAccessibleForFree":true,"license":"https:\/\/creativecommons.org\/licenses\/by-nc-nd\/3.0\/legalcode","copyrightHolder":[{"@type":"Person","name":"Kawarabayashi, Ken-ichi","givenName":"Ken-ichi","familyName":"Kawarabayashi"},{"@type":"Person","name":"Kobayashi, Yusuke","givenName":"Yusuke","familyName":"Kobayashi"}],"copyrightYear":"2012","accessMode":"textual","accessModeSufficient":"textual","creativeWorkStatus":"Published","inLanguage":"en-US","sameAs":"https:\/\/doi.org\/10.4230\/LIPIcs.STACS.2012.206","publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","isPartOf":{"@type":"PublicationVolume","@id":"#volume6217","volumeNumber":14,"name":"29th International Symposium on Theoretical Aspects of Computer Science (STACS 2012)","dateCreated":"2012-02-24","datePublished":"2012-02-24","editor":[{"@type":"Person","name":"D\u00fcrr, Christoph","givenName":"Christoph","familyName":"D\u00fcrr"},{"@type":"Person","name":"Wilke, Thomas","givenName":"Thomas","familyName":"Wilke"}],"isAccessibleForFree":true,"publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","hasPart":"#article7070","isPartOf":{"@type":"Periodical","@id":"#series116","name":"Leibniz International Proceedings in Informatics","issn":"1868-8969","isAccessibleForFree":true,"publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","hasPart":"#volume6217"}}}