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Improved bounds for hypohamiltonian graphs

Jan Goedgebeur (UGent) and Carol Zamfirescu (UGent)
(2017) ARS MATHEMATICA CONTEMPORANEA. 13(2). p.235-257
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Organization
Abstract
A graph G is hypohamiltonian if G is non-hamiltonian and G - nu is hamiltonian for every nu is an element of V (G). In the following, every graph is assumed to be hypohamiltonian. Aldred, Wormald, and McKay gave a list of all graphs of order at most 17. In this article, we present an algorithm to generate all graphs of a given order and apply it to prove that there exist exactly 14 graphs of order 18 and 34 graphs of order 19. We also extend their results in the cubic case. Furthermore, we show that (i) the smallest graph of girth 6 has order 25, (ii) the smallest planar graph has order at least 23, (iii) the smallest cubic planar graph has order at least 54, and (iv) the smallest cubic planar graph of girth 5 with non-trivial automorphism group has order 78.
Keywords
Hamiltonian, hypohamiltonian, planar, girth, cubic graph, exhaustive generation, CUBIC PLANAR GRAPHS, HYPOTRACEABLE GRAPHS, FAST GENERATION, VERTICES

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Citation

Please use this url to cite or link to this publication:

Chicago
Goedgebeur, Jan, and Carol Zamfirescu. 2017. “Improved Bounds for Hypohamiltonian Graphs.” Ars Mathematica Contemporanea 13 (2): 235–257.
APA
Goedgebeur, J., & Zamfirescu, C. (2017). Improved bounds for hypohamiltonian graphs. ARS MATHEMATICA CONTEMPORANEA, 13(2), 235–257.
Vancouver
1.
Goedgebeur J, Zamfirescu C. Improved bounds for hypohamiltonian graphs. ARS MATHEMATICA CONTEMPORANEA. 2017;13(2):235–57.
MLA
Goedgebeur, Jan, and Carol Zamfirescu. “Improved Bounds for Hypohamiltonian Graphs.” ARS MATHEMATICA CONTEMPORANEA 13.2 (2017): 235–257. Print.
@article{8512918,
  abstract     = {A graph G is hypohamiltonian if G is non-hamiltonian and G - nu is hamiltonian for every nu is an element of V (G). In the following, every graph is assumed to be hypohamiltonian. Aldred, Wormald, and McKay gave a list of all graphs of order at most 17. In this article, we present an algorithm to generate all graphs of a given order and apply it to prove that there exist exactly 14 graphs of order 18 and 34 graphs of order 19. We also extend their results in the cubic case. Furthermore, we show that (i) the smallest graph of girth 6 has order 25, (ii) the smallest planar graph has order at least 23, (iii) the smallest cubic planar graph has order at least 54, and (iv) the smallest cubic planar graph of girth 5 with non-trivial automorphism group has order 78.},
  author       = {Goedgebeur, Jan and Zamfirescu, Carol},
  issn         = {1855-3966},
  journal      = {ARS MATHEMATICA CONTEMPORANEA},
  keywords     = {Hamiltonian,hypohamiltonian,planar,girth,cubic graph,exhaustive generation,CUBIC PLANAR GRAPHS,HYPOTRACEABLE GRAPHS,FAST GENERATION,VERTICES},
  language     = {eng},
  number       = {2},
  pages        = {235--257},
  title        = {Improved bounds for hypohamiltonian graphs},
  volume       = {13},
  year         = {2017},
}

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