Non-Hamiltonian simple 3-polytopes having just two types of faces

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It is shown that for every value of an integer k, k≥11, there exist 3-valent 3-connected planar graphs having just two types of faces, pentagons and k-gons, and which are non- Hamiltonian. Moreover, there exist ε{lunate}=ε{lunate}(k) > 0, for these values of k, and sequences (Gn)n=1 of the said graphs for which V(Gn)→∞ and the size of a largest circuit of Gn is at most (1-ε{lunate})V(Gn); similar result for the size of a largest path in such graphs is established for all k, k≥11, except possibly for k = 14, 17, 22 and k = 5m+ 5 for all m≥2.

Original languageEnglish
Pages (from-to)87-101
Number of pages15
JournalDiscrete Mathematics
Issue number1
StatePublished - 1980

Bibliographical note

Funding Information:
* Inspired by a conversation with D.H. Younger, while the author was visiting J.A. Bondy at the University of Waterloo, June, 1978; extended while the author visited A. Kotzig at the C.R.M., University of Montreal. “f Res.2arclr supported by NRC of Canada Grant A7331.

ASJC Scopus subject areas

  • Theoretical Computer Science
  • Discrete Mathematics and Combinatorics


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