Potential forbidden triples implying hamiltonicity: for sufficiently large graphs

Ralph J. Faudree; Ronald J. Gould; Michael S. Jacobson

Discussiones Mathematicae Graph Theory (2005)

  • Volume: 25, Issue: 3, page 273-289
  • ISSN: 2083-5892

Abstract

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In [1], Brousek characterizes all triples of connected graphs, G₁,G₂,G₃, with G i = K 1 , 3 for some i = 1,2, or 3, such that all G₁G₂ G₃-free graphs contain a hamiltonian cycle. In [8], Faudree, Gould, Jacobson and Lesniak consider the problem of finding triples of graphs G₁,G₂,G₃, none of which is a K 1 , s , s ≥ 3 such that G₁G₂G₃-free graphs of sufficiently large order contain a hamiltonian cycle. In [6], a characterization was given of all triples G₁,G₂,G₃ with none being K 1 , 3 , such that all G₁G₂G₃-free graphs are hamiltonian. This result, together with the triples given by Brousek, completely characterize the forbidden triples G₁,G₂,G₃ such that all G₁G₂G₃-free graphs are hamiltonian. In this paper we consider the question of which triples (including K 1 , s , s ≥ 3) of forbidden subgraphs potentially imply all sufficiently large graphs are hamiltonian. For s ≥ 4 we characterize these families.

How to cite

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Ralph J. Faudree, Ronald J. Gould, and Michael S. Jacobson. "Potential forbidden triples implying hamiltonicity: for sufficiently large graphs." Discussiones Mathematicae Graph Theory 25.3 (2005): 273-289. <http://eudml.org/doc/270266>.

@article{RalphJ2005,
abstract = {In [1], Brousek characterizes all triples of connected graphs, G₁,G₂,G₃, with $G_i = K_\{1,3\}$ for some i = 1,2, or 3, such that all G₁G₂ G₃-free graphs contain a hamiltonian cycle. In [8], Faudree, Gould, Jacobson and Lesniak consider the problem of finding triples of graphs G₁,G₂,G₃, none of which is a $K_\{1,s\}$, s ≥ 3 such that G₁G₂G₃-free graphs of sufficiently large order contain a hamiltonian cycle. In [6], a characterization was given of all triples G₁,G₂,G₃ with none being $K_\{1,3\}$, such that all G₁G₂G₃-free graphs are hamiltonian. This result, together with the triples given by Brousek, completely characterize the forbidden triples G₁,G₂,G₃ such that all G₁G₂G₃-free graphs are hamiltonian. In this paper we consider the question of which triples (including $K_\{1,s\}$, s ≥ 3) of forbidden subgraphs potentially imply all sufficiently large graphs are hamiltonian. For s ≥ 4 we characterize these families.},
author = {Ralph J. Faudree, Ronald J. Gould, Michael S. Jacobson},
journal = {Discussiones Mathematicae Graph Theory},
keywords = {hamiltonian; forbidden subgraph; claw-free; induced subgraph; hamiltonian graph; forbidden triples},
language = {eng},
number = {3},
pages = {273-289},
title = {Potential forbidden triples implying hamiltonicity: for sufficiently large graphs},
url = {http://eudml.org/doc/270266},
volume = {25},
year = {2005},
}

TY - JOUR
AU - Ralph J. Faudree
AU - Ronald J. Gould
AU - Michael S. Jacobson
TI - Potential forbidden triples implying hamiltonicity: for sufficiently large graphs
JO - Discussiones Mathematicae Graph Theory
PY - 2005
VL - 25
IS - 3
SP - 273
EP - 289
AB - In [1], Brousek characterizes all triples of connected graphs, G₁,G₂,G₃, with $G_i = K_{1,3}$ for some i = 1,2, or 3, such that all G₁G₂ G₃-free graphs contain a hamiltonian cycle. In [8], Faudree, Gould, Jacobson and Lesniak consider the problem of finding triples of graphs G₁,G₂,G₃, none of which is a $K_{1,s}$, s ≥ 3 such that G₁G₂G₃-free graphs of sufficiently large order contain a hamiltonian cycle. In [6], a characterization was given of all triples G₁,G₂,G₃ with none being $K_{1,3}$, such that all G₁G₂G₃-free graphs are hamiltonian. This result, together with the triples given by Brousek, completely characterize the forbidden triples G₁,G₂,G₃ such that all G₁G₂G₃-free graphs are hamiltonian. In this paper we consider the question of which triples (including $K_{1,s}$, s ≥ 3) of forbidden subgraphs potentially imply all sufficiently large graphs are hamiltonian. For s ≥ 4 we characterize these families.
LA - eng
KW - hamiltonian; forbidden subgraph; claw-free; induced subgraph; hamiltonian graph; forbidden triples
UR - http://eudml.org/doc/270266
ER -

References

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  1. [1] P. Bedrossian, Forbidden subgraph and minimum degree conditions for hamiltonicity (Ph.D. Thesis, Memphis State University, 1991). 
  2. [2] J. Brousek, Forbidden triples and hamiltonicity, Discrete Math. 251 (2002) 71-76, doi: 10.1016/S0012-365X(01)00326-0. Zbl1002.05044
  3. [3] J. Brousek, Z. Ryjácek and I. Schiermeyer, Forbidden subgraphs, stability and hamiltonicity, 18th British Combinatorial Conference (London, 1997), Discrete Math. 197/198 (1999) 143-155, doi: 10.1016/S0012-365X(98)00229-5. 
  4. [4] G. Chartrand and L. Lesniak, Graphs & Digraphs (3rd Edition, Chapman & Hall, 1996). 
  5. [5] R.J. Faudree and R.J. Gould, Characterizing forbidden pairs for hamiltonian properties, Discrete Math. 173 (1997) 45-60, doi: 10.1016/S0012-365X(96)00147-1. Zbl0879.05050
  6. [6] R.J. Faudree, R.J. Gould and M.S. Jacobson, Forbidden triples implying hamiltonicity: for all graphs, Discuss. Math. Graph Theory 24 (2004) 47-54, doi: 10.7151/dmgt.1212. Zbl1060.05063
  7. [7] R.J. Faudree, R.J. Gould and M.S. Jacobson, Forbidden triples including K 1 , 3 implying hamiltonicity: for sufficiently large graphs, preprint. Zbl1143.05051
  8. [8] R.J. Faudree, R.J. Gould, M.S. Jacobson and L. Lesniak, Characterizing forbidden clawless triples implying hamiltonian graphs, Discrete Math. 249 (2002) 71-81, doi: 10.1016/S0012-365X(01)00235-7. Zbl0990.05091

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