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PreprintCC BY 4.0arXiv

A tris of perfect matchings in bridgeless claw-free cubic graphs

  • Jean Paul Zerafa
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Abstract

A proof of the cycle double cover conjecture was recently announced, yielding an $8$-cycle double cover for every bridgeless graph. The stronger $5$-cycle double cover conjecture, which is still open, is equivalent to the statement that the edge set of every bridgeless claw-free cubic graph can be covered by at most four perfect matchings. Perfect matchings in bridgeless cubic graphs have been studied extensively, with two of the main conjectures in this area being the Berge--Fulkerson and the Fan--Raspaud conjectures. The latter, a consequence of the former, states that every bridgeless cubic graph admits three perfect matchings M1, M2, M3 such that M1∩ M2∩ M3=∅. Here we show that the Fan--Raspaud conjecture is true for bridgeless claw-free cubic graphs. This also gives further information on the interaction of perfect matchings in a class where the $5$-cycle double cover conjecture requires control of four of them.

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Cite

BibTeXRIS
BibTeX
@misc{zerafa2026tris,
  title = {{A tris of perfect matchings in bridgeless claw-free cubic graphs}},
  author = {Jean Paul Zerafa},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.28118},
  eprint = {2609.28118},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.28118},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - A tris of perfect matchings in bridgeless claw-free cubic graphs
AU  - Jean Paul Zerafa
PY  - 2026
DA  - 2026/09/23
PB  - arXiv
DO  - 10.48550/arxiv.2609.28118
UR  - https://arxiv.org/abs/2609.28118
AB  - A proof of the cycle double cover conjecture was recently announced, yielding an $8$-cycle double cover for every bridgeless graph. The stronger $5$-cycle double cover conjecture, which is still open, is equivalent to the statement that the edge set of every bridgeless claw-free cubic graph can be covered by at most four perfect matchings. Perfect matchings in bridgeless cubic graphs have been studied extensively, with two of the main conjectures in this area being the Berge--Fulkerson and the Fan--Raspaud conjectures. The latter, a consequence of the former, states that every bridgeless cubic graph admits three perfect matchings M_1, M_2, M_3 such that M_1∩ M_2∩ M_3=∅. Here we show that the Fan--Raspaud conjecture is true for bridgeless claw-free cubic graphs. This also gives further information on the interaction of perfect matchings in a class where the $5$-cycle double cover conjecture requires control of four of them.
ER  -