Periodica Mathematica Hungarica

, Volume 14, Issue 1, pp 57–68 | Cite as

Smallest maximally nonhamiltonian graphs

  • L. Clark
  • R. Entringer


A graphG ismaximally nonhamiltonian iffG is not hamiltonian butG + e is hamiltonian for each edgee inGc, i.e., any two non-adjacent vertices ofG are ends of a hamiltonian path. Bollobás posed the problem of finding the least number of edges,f(n), possible in a maximally nonhamiltonian graph of ordern. Results of Bondy show thatf(n)3/2n forn ≤ 7. We exhibit graphs of even ordern ≥ 36 for which the bound is attained. These graphs are the “snarks”,Jk, of Isaacs and mild variations of them. For oddn ≥ 55 we construct graphs from the graphsJk showing that in this case,f(n) = 3n + 1/2 or 3n + 3/2 and leave the determination of which is correct as an open problem. Finally we note that the graphsJk, k ≤ 7 are hypohamiltonian cubics with girth 6.

AMS (MOS) subject classifications (1980)

Primary 05C35 Secondary 05C45 

Key words and phrases

Graphs Hamiltonian 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    B. Bollobás,Extremal graph theory, Academic Press, London, 1978.MR 80a: 05120Google Scholar
  2. [2]
    J. A. Bondy, Variations on the hamiltonian theme,Canad. Math. Bull. 15 (1972), 57–62.MR 47 # 3241Google Scholar
  3. [3]
    J. A. Bondy andU. S. R. Murty,Graph theory with applications, American Elsevier, New York, 1976.MR 54 # 117Google Scholar
  4. [4]
    F. C. Bussemaker, S. Čobeljić, D. M. Cvetković andJ. J. Seidel,Computer investigation of cubic graphs (Report 76-wsk-01), Technological University, Eindhoven, The Netherlands, January 1976.Google Scholar
  5. [5]
    V. Chvátal, Flip-flops in hypo-Hamiltonian graphs,Canad. Math. Bull. 16 (1973), 33–41.MR 51 # 7939Google Scholar
  6. [6]
    L. Clark, Ph. D. Dissertation, University of New Mexico, Albuquerque, May 1980.Google Scholar
  7. [7]
    R. Isaacs, Infinite families of nontrivial trivalent graphs which are not Tait colorable,Amer. Math. Monthly 82 (1975), 221–239.MR 52 # 2940Google Scholar
  8. [8]
    O. Ore, Arc coverings of graphs,Ann. Mat. Pura Appl. 55 (1961), 315–321.MR 28 # 5441Google Scholar
  9. [9]
    W. T. Tutte, A non-Hamiltonian graph,Canad. Math. Bull. 3 (1960), 1–5.MR 22 # 4646Google Scholar

Copyright information

© Akadémiai Kiadó 1983

Authors and Affiliations

  • L. Clark
    • 1
  • R. Entringer
    • 1
  1. 1.Department of MathematicsUniversity of New MexicoAlbuquerqueUSA

Personalised recommendations