Disjoint odd circuits in a bridgeless cubic graph can be quelled by a single perfect matching - Archive ouverte HAL Accéder directement au contenu
Article Dans Une Revue Journal of Combinatorial Theory, Series B Année : 2023

Disjoint odd circuits in a bridgeless cubic graph can be quelled by a single perfect matching

Résumé

Let G be a bridgeless cubic graph. The Berge-Fulkerson Conjecture (1970s) states that G admits a list of six perfect matchings such that each edge of G belongs to exactly two of these perfect matchings. If answered in the affirmative, two other recent conjectures would also be true: the Fan-Raspaud Conjecture (1994), which states that G admits three perfect matchings such that every edge of G belongs to at most two of them; and a conjecture by Mazzuoccolo (2013), which states that G admits two perfect matchings whose deletion yields a bipartite subgraph of G. It can be shown that given an arbitrary perfect matching of G, it is not always possible to extend it to a list of three or six perfect matchings satisfying the statements of the Fan-Raspaud and the Berge-Fulkerson conjectures, respectively. In this paper, we show that given any 1 +-factor F (a spanning subgraph of G such that its vertices have degree at least 1) and an arbitrary edge e of G, there always exists a perfect matching M of G containing e such that G \ (F ∪ M) is bipartite. Our result implies Mazzuoccolo's conjecture, but not only. It also implies that given any collection of disjoint odd circuits in G, there exists a perfect of G containing at least one edge of each circuit in this collection.
Fichier principal
Vignette du fichier
Bipartite_Complement.pdf (223.48 Ko) Télécharger le fichier
Origine : Fichiers produits par l'(les) auteur(s)

Dates et versions

hal-04019264 , version 1 (08-03-2023)

Identifiants

Citer

František Kardoš, Edita Máčajová, Jean Paul Zerafa. Disjoint odd circuits in a bridgeless cubic graph can be quelled by a single perfect matching. Journal of Combinatorial Theory, Series B, 2023, 160, pp.1-14. ⟨10.1016/j.jctb.2022.12.003⟩. ⟨hal-04019264⟩

Collections

CNRS TDS-MACS
8 Consultations
20 Téléchargements

Altmetric

Partager

Gmail Facebook X LinkedIn More