Cores, joins and the Fano-flow conjectures

TL;DR

This paper explores the relationships between cores, joins, and Fano-flow conjectures in bridgeless cubic graphs, proving equivalences, disproving some conjectures, and establishing bounds on oddness and weak oddness.

Contribution

It establishes equivalences between key conjectures, disproves a proposed conjecture, and introduces bounds on graph oddness based on core structures.

Findings

Proves the equivalence of the Fan-Raspaud Conjecture with a conjecture on cores.

Disproves a conjecture on perfect matchings in cubic graphs.

Provides upper bounds on weak oddness and oddness using core concepts.

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 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.