Cores, joins and the Fano-flow conjectures
Ligang Jin, Giuseppe Mazzuoccolo, Eckhard Steffen · Discussiones Mathematicae Graph Theory · 2017
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.