Girth Six Cubic Graphs Have Petersen Minors


We prove that every 3-regular graph with no circuit of length less than six has a subgraph isomorphic to a subdivision of the Petersen graph.

This is a preview of subscription content, access via your institution.


  1. [1]

    B. Alspach, L. Goddyn and C.-Q. Zhang: Graphs with the circuit cover property, Trans. Amer. Math. Soc.344 (1994), 131–154.

    MathSciNet  Article  Google Scholar 

  2. [2]

    A. Huck: Reducible configurations for the cycle double cover conjecture, Proceedings of the 5th Twente Workshop on Graphs and Combinatorial Optimization (Enschede, 1997), Discrete Appl. Math.99 (2000), 71–90.

    MathSciNet  Article  Google Scholar 

  3. [3]

    M. Kochol: Cubic graphs without a Petersen minor have nowhere-zero 5-flows, Acta Math. Univ. Comenian. (N.S.)68 (1999), 249–252.

    MathSciNet  MATH  Google Scholar 

  4. [4]

    N. Robertson, P. D. Seymour and R. Thomas: Cyclically 5-connected cubic graphs, J. Combin. Theory Ser. B125 (2017), 132–167.

    MathSciNet  Article  Google Scholar 

  5. [5]

    N. Robertson, P. D. Seymour and R. Thomas: Excluded minors in cubic graphs, J. Combin. Theory Ser. B138 (2019), 219–285.

    MathSciNet  Article  Google Scholar 

Download references

Author information



Corresponding authors

Correspondence to Neil Robertson or P. D. Seymour or Robin Thomas.

Additional information

Research partially performed under a consulting agreement with Bellcore, 445 South St., Morristown, New Jersey 07960, USA, and partially supported by DIMACS Center, Rutgers University, New Brunswick, New Jersey 08903, USA.

Partially supported by NSF under Grant No. DMS-9401981 and by ONR under Grant No. N00014-92-J-1965.

Research partially performed while Seymour was employed at Bellcore, and partially supported by ONR grant N00014-10-1-0680, and NSF grants DMS-1265563 and DMS-1800053.

Partially supported by NSF under Grants No. DMS-9623031, DMS-1202640 and DMS- 1700157 and by ONR under Contract No. N00014-93-1-0325.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Robertson, N., Seymour, P.D. & Thomas, R. Girth Six Cubic Graphs Have Petersen Minors. Combinatorica 39, 1413–1423 (2019).

Download citation

Mathematics Subject Classification (2010)

  • 05C75
  • 05C83