Abstract
We describe a sufficient condition for graphs used in a construction of Thomassen (which yields hypohamiltonian graphs) to produce maximally non-hamiltonian (MNH) graphs as well. Then we show that the Coxeter graph fulfils this sufficient condition, and thus applying the Thomassen's construction to multiple copies of the Coxeter graph yields infinitely many MNH graphs with girth 7. So far, the Coxeter graph was the only known example of a MNH graph of girth 7; also no MNH graph of girth greater than 7 has been found yet. Finally, the Isaacs' flower snarksJ k for oddk ≥ 5 are shown to fulfil (for certain vertices) this sufficient condition as well.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Clark, L., Entringer, R.: Smallest maximally nonhamiltonian graphs, Period. Math. Hung.14, 57–68 (1983)
Horák, P., Širáň, J.: On a construction of Thomassen, Graphs and Combinatorics2, 347–350 (1986)
Isaacs, R.: Infinite families of nontrivial trivalent graphs that are not Tait colorable, Amer. Math. Mon.82, 221–239 (1975)
Thomassen, C.: Hypohamiltonian and hypotraceable graphs, Discrete Math.9, 91–96 (1974)
Author information
Authors and Affiliations
Additional information
The research of author was partially supported by Grant No. 2/1138/94 “Computational models, algorithms and complexity” of Slovak Academy of Sciences and by EC Cooperative Action IC1000 “Algorithms for Future Technologies” (Project ALTEC)
Rights and permissions
About this article
Cite this article
Stacho, L. Maximally non-hamiltonian graphs of girth 7. Graphs and Combinatorics 12, 361–371 (1996). https://doi.org/10.1007/BF01858469
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/BF01858469