# Cores, joins and the Fano-flow conjectures

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Ligang Jin

Paderborn University

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Giuseppe Mazzuoccolo

Universit\`{a} di Modena e Reggio Emilia

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Eckhard Steffen

Paderborn University

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**Minisymposium:**
GENERAL SESSION TALKS

**Content:**
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 talk, 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 is the minimum number of odd components in the complement of a join of this graph.
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.