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On Fulkerson conjecture

Jean-Luc Fouquet; Jean-Marie Vanherpe

Discussiones Mathematicae Graph Theory (2011)

  • Volume: 31, Issue: 2, page 253-272
  • ISSN: 2083-5892

Abstract

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If G is a bridgeless cubic graph, Fulkerson conjectured that we can find 6 perfect matchings (a Fulkerson covering) with the property that every edge of G is contained in exactly two of them. A consequence of the Fulkerson conjecture would be that every bridgeless cubic graph has 3 perfect matchings with empty intersection (this problem is known as the Fan Raspaud Conjecture). A FR-triple is a set of 3 such perfect matchings. We show here how to derive a Fulkerson covering from two FR-triples. Moreover, we give a simple proof that the Fulkerson conjecture holds true for some classes of well known snarks.

How to cite

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Jean-Luc Fouquet, and Jean-Marie Vanherpe. "On Fulkerson conjecture." Discussiones Mathematicae Graph Theory 31.2 (2011): 253-272. <http://eudml.org/doc/270931>.

@article{Jean2011,
abstract = {If G is a bridgeless cubic graph, Fulkerson conjectured that we can find 6 perfect matchings (a Fulkerson covering) with the property that every edge of G is contained in exactly two of them. A consequence of the Fulkerson conjecture would be that every bridgeless cubic graph has 3 perfect matchings with empty intersection (this problem is known as the Fan Raspaud Conjecture). A FR-triple is a set of 3 such perfect matchings. We show here how to derive a Fulkerson covering from two FR-triples. Moreover, we give a simple proof that the Fulkerson conjecture holds true for some classes of well known snarks.},
author = {Jean-Luc Fouquet, Jean-Marie Vanherpe},
journal = {Discussiones Mathematicae Graph Theory},
keywords = {cubic graph; perfect matchings; Fulkerson covering; Fan-Raspaud triple; FR-triple},
language = {eng},
number = {2},
pages = {253-272},
title = {On Fulkerson conjecture},
url = {http://eudml.org/doc/270931},
volume = {31},
year = {2011},
}

TY - JOUR
AU - Jean-Luc Fouquet
AU - Jean-Marie Vanherpe
TI - On Fulkerson conjecture
JO - Discussiones Mathematicae Graph Theory
PY - 2011
VL - 31
IS - 2
SP - 253
EP - 272
AB - If G is a bridgeless cubic graph, Fulkerson conjectured that we can find 6 perfect matchings (a Fulkerson covering) with the property that every edge of G is contained in exactly two of them. A consequence of the Fulkerson conjecture would be that every bridgeless cubic graph has 3 perfect matchings with empty intersection (this problem is known as the Fan Raspaud Conjecture). A FR-triple is a set of 3 such perfect matchings. We show here how to derive a Fulkerson covering from two FR-triples. Moreover, we give a simple proof that the Fulkerson conjecture holds true for some classes of well known snarks.
LA - eng
KW - cubic graph; perfect matchings; Fulkerson covering; Fan-Raspaud triple; FR-triple
UR - http://eudml.org/doc/270931
ER -

References

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  2. [2] G. Fan and A. Raspaud, Fulkerson's Conjecture and Circuit Covers, J. Combin. Theory (B) 61 (1994) 133-138, doi: 10.1006/jctb.1994.1039. Zbl0811.05053
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  5. [5] M.K. Goldberg, Construction of class 2 graphs with maximum vertex degree 3, J. Combin. Theory (B) 31 (1981) 282-291, doi: 10.1016/0095-8956(81)90030-7. Zbl0449.05037
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  7. [7] R. Rizzi, Indecomposable r-graphs and some other counterexamples, J. Graph Theory 32 (1999) 1-15, doi: 10.1002/(SICI)1097-0118(199909)32:1<1::AID-JGT1>3.0.CO;2-B Zbl0932.05072
  8. [8] Z. Skupień Exponentially many hypohamiltonian graphs, Graphs, Hypergraphs and Matroids III (M. Borowiecki and Z. Skupień, eds.) Proc. Conf. Kalsk, 1988, Higher College of Enginering, Zielona Góra, 1989, pp. 123-132. 
  9. [9] J.J. Watkins, Snarks, Graph Theory and Its Applications: East and West (Jinan, 1986), Ann. New York Acad. Sci. vol. 576, New York Acad. Sci. (New York, 1989) pp. 606-622. 
  10. [10] R. Hao, J. Niu, X. Wang, C.-Q. Zhang and T. Zhang, A note on Berge-Fulkerson coloring, Discrete Math. 309 (2009) 4235-4240, doi: 10.1016/j.disc.2008.12.024. Zbl1209.05084

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