Graphs and Combinatorics

, Volume 3, Issue 1, pp 299–311 | Cite as

Cycles through five edges in 3-connected cubic graphs

  • R. E. L. Aldred
  • D. A. Holton


Given five edges in a 3-connected cubic graph there are obvious reasons why there may not be one cycle passing through all of them. For instance, an odd subset of the edges may form a cutset of the graph. By restricting the sets of five edges in a natural way we are able to give necessary and sufficient conditions for the set to be a subset of edges of some cycle. It follows as a corollary that, under suitable restrictions, any five edges of a cyclically 5-edge connected cubic graph lie on a cycle.


Obvious Reason Suitable Restriction Cycle Passing 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Aldred, R.E.L., Holton, D.A., Thomassen, C.: Cycles through four edges in 3-connected cubic graphs. Graphs and Combinatorics1, 1–5 (1985)Google Scholar
  2. 2.
    Bussemaker, F.C., Cobeljic, S., Cvetkovic, D.M., Seidel, J.J.: Computer investigation of cubic graphs. Report 76-WSK-01. Eindhoven: Technological University 1976Google Scholar
  3. 3.
    Ellingham, M.N.: Cycles in 3-connected cubic graphs. M. Sc. Thesis. University of Melbourne 1982Google Scholar
  4. 4.
    Ellingham, M.N., Holton, D.A., Little, C.H.C.: Cycles through ten vertices in 3-connected cubic graphs. Combinatorica4, 256–273 (1984)Google Scholar
  5. 5.
    Häggvist, R., Thomassen, C.: Circuits through specified edges. Discrete Math.41, 29–34 (1982)Google Scholar
  6. 6.
    Holton, D.A., McKay, B.D., Plummer, M.D., Thomassen, C.: A nine point theorem for 3-connected graphs. Combinatorica2, 53–62 (1982)Google Scholar
  7. 7.
    Holton, D.A., Thomassen, C.: Research Problem. Discrete Math. (to appear)Google Scholar
  8. 8.
    Lovàsz, L.: Problem 5. Period. Math. Hung.4, 82 (1974)Google Scholar
  9. 9.
    Thomassen, C.: Girth in graphs. J. Comb. Theory (B)35, 129–141 (1983)Google Scholar

Copyright information

© Springer-Verlag 1987

Authors and Affiliations

  • R. E. L. Aldred
    • 1
  • D. A. Holton
    • 1
  1. 1.Department of Mathematics and StatisticsUniversity of OtagoDunedinNew Zealand

Personalised recommendations