Periodica Mathematica Hungarica

, Volume 14, Issue 1, pp 57–68

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 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Copyright information

© Akadémiai Kiadó 1983

Authors and Affiliations

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

Personalised recommendations