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

Joseph Zaks

doi:10.1016/0012-365X(90)90289-T

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

Joseph Zaks

Department of Mathematics

Research output: Contribution to journal › Article › peer-review

Abstract

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 (G_n)^∞_n=1 of the said graphs for which V(G_n)→∞ and the size of a largest circuit of G_n is at most (1-ε{lunate})V(G_n); 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 language	English
Pages (from-to)	87-101
Number of pages	15
Journal	Discrete Mathematics
Volume	29
Issue number	1
DOIs	https://doi.org/10.1016/0012-365X(90)90289-T
State	Published - 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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10.1016/0012-365X(90)90289-T

Cite this

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title = "Non-Hamiltonian simple 3-polytopes having just two types of faces",

abstract = "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.",

author = "Joseph Zaks",

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.",

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AU - Zaks, Joseph

N1 - 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.

PY - 1980

Y1 - 1980

N2 - 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.

AB - 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.

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DO - 10.1016/0012-365X(90)90289-T

M3 - Article

AN - SCOPUS:0007270137

SN - 0012-365X

VL - 29

SP - 87

EP - 101

JO - Discrete Mathematics

JF - Discrete Mathematics

IS - 1

ER -

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

Abstract

Bibliographical note

ASJC Scopus subject areas

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