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.
@article{bwmeta1.element.doi-10_7151_dmgt_1999,
author = {Ligang Jin and Eckhard Steffen and Giuseppe Mazzuoccolo},
title = {Cores, Joins and the Fano-Flow Conjectures},
journal = {Discussiones Mathematicae Graph Theory},
volume = {38},
year = {2018},
pages = {165-175},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.doi-10_7151_dmgt_1999}
}
Ligang Jin; Eckhard Steffen; Giuseppe Mazzuoccolo. Cores, Joins and the Fano-Flow Conjectures. Discussiones Mathematicae Graph Theory, Tome 38 (2018) pp. 165-175. http://gdmltest.u-ga.fr/item/bwmeta1.element.doi-10_7151_dmgt_1999/