Periodica Mathematica Hungarica

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

Smallest maximally nonhamiltonian graphs

  • L. Clark
  • R. Entringer
Article

Abstract

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 

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References

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Copyright information

© Akadémiai Kiadó 1983

Authors and Affiliations

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

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