License: Creative Commons Attribution-NonCommercial-NoDerivs 3.0 Unported license (CC BY-NC-ND 3.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.STACS.2012.206
URN: urn:nbn:de:0030-drops-34173
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2012/3417/
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Kawarabayashi, Ken-ichi ; Kobayashi, Yusuke

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

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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.

BibTeX - Entry

@InProceedings{kawarabayashi_et_al:LIPIcs:2012:3417,
  author =	{Ken-ichi Kawarabayashi and Yusuke Kobayashi},
  title =	{{Edge-disjoint Odd Cycles in 4-edge-connected Graphs}},
  booktitle =	{29th International Symposium on Theoretical Aspects of Computer Science (STACS 2012)},
  pages =	{206--217},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-939897-35-4},
  ISSN =	{1868-8969},
  year =	{2012},
  volume =	{14},
  editor =	{Christoph D{\"u}rr and Thomas Wilke},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{http://drops.dagstuhl.de/opus/volltexte/2012/3417},
  URN =		{urn:nbn:de:0030-drops-34173},
  doi =		{10.4230/LIPIcs.STACS.2012.206},
  annote =	{Keywords: odd-cycles, disjoint paths problem, Erd{\"o}s-Posa property, packing algorithm, 4-edge-connectivity }
}

Keywords: odd-cycles, disjoint paths problem, Erdös-Posa property, packing algorithm, 4-edge-connectivity
Collection: 29th International Symposium on Theoretical Aspects of Computer Science (STACS 2012)
Issue Date: 2012
Date of publication: 24.02.2012

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