# Cores, Joins and the Fano-Flow Conjectures

Ligang Jin; Eckhard Steffen; Giuseppe Mazzuoccolo

Discussiones Mathematicae Graph Theory (2018)

- Volume: 38, Issue: 1, page 165-175
- ISSN: 2083-5892

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topLigang Jin, Eckhard Steffen, and Giuseppe Mazzuoccolo. "Cores, Joins and the Fano-Flow Conjectures." Discussiones Mathematicae Graph Theory 38.1 (2018): 165-175. <http://eudml.org/doc/288523>.

@article{LigangJin2018,

abstract = {The Fan-Raspaud Conjecture states that every bridgeless cubic graph has three 1-factors with empty intersection. A weaker one than this conjecture is that every bridgeless cubic graph has two 1-factors and one join with empty intersection. Both of these two conjectures can be related to conjectures on Fano-flows. In this paper, we show that these two conjectures are equivalent to some statements on cores and weak cores of a bridgeless cubic graph. In particular, we prove that the Fan-Raspaud Conjecture is equivalent to a conjecture proposed in [E. Steffen, 1-factor and cycle covers of cubic graphs, J. Graph Theory 78 (2015) 195–206]. Furthermore, we disprove a conjecture proposed in [G. Mazzuoccolo, New conjectures on perfect matchings in cubic graphs, Electron. Notes Discrete Math. 40 (2013) 235–238] and we propose a new version of it under a stronger connectivity assumption. The weak oddness of a cubic graph G is the minimum number of odd components (i.e., with an odd number of vertices) in the complement of a join of G. We obtain an upper bound of weak oddness in terms of weak cores, and thus an upper bound of oddness in terms of cores as a by-product.},

author = {Ligang Jin, Eckhard Steffen, Giuseppe Mazzuoccolo},

journal = {Discussiones Mathematicae Graph Theory},

keywords = {cubic graphs; Fan-Raspaud Conjecture; cores; weak-cores},

language = {eng},

number = {1},

pages = {165-175},

title = {Cores, Joins and the Fano-Flow Conjectures},

url = {http://eudml.org/doc/288523},

volume = {38},

year = {2018},

}

TY - JOUR

AU - Ligang Jin

AU - Eckhard Steffen

AU - Giuseppe Mazzuoccolo

TI - Cores, Joins and the Fano-Flow Conjectures

JO - Discussiones Mathematicae Graph Theory

PY - 2018

VL - 38

IS - 1

SP - 165

EP - 175

AB - The Fan-Raspaud Conjecture states that every bridgeless cubic graph has three 1-factors with empty intersection. A weaker one than this conjecture is that every bridgeless cubic graph has two 1-factors and one join with empty intersection. Both of these two conjectures can be related to conjectures on Fano-flows. In this paper, we show that these two conjectures are equivalent to some statements on cores and weak cores of a bridgeless cubic graph. In particular, we prove that the Fan-Raspaud Conjecture is equivalent to a conjecture proposed in [E. Steffen, 1-factor and cycle covers of cubic graphs, J. Graph Theory 78 (2015) 195–206]. Furthermore, we disprove a conjecture proposed in [G. Mazzuoccolo, New conjectures on perfect matchings in cubic graphs, Electron. Notes Discrete Math. 40 (2013) 235–238] and we propose a new version of it under a stronger connectivity assumption. The weak oddness of a cubic graph G is the minimum number of odd components (i.e., with an odd number of vertices) in the complement of a join of G. We obtain an upper bound of weak oddness in terms of weak cores, and thus an upper bound of oddness in terms of cores as a by-product.

LA - eng

KW - cubic graphs; Fan-Raspaud Conjecture; cores; weak-cores

UR - http://eudml.org/doc/288523

ER -