# 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

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topRalph 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

top- [1] P. Bedrossian, Forbidden subgraph and minimum degree conditions for hamiltonicity (Ph.D. Thesis, Memphis State University, 1991).
- [2] J. Brousek, Forbidden triples and hamiltonicity, Discrete Math. 251 (2002) 71-76, doi: 10.1016/S0012-365X(01)00326-0. Zbl1002.05044
- [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] G. Chartrand and L. Lesniak, Graphs & Digraphs (3rd Edition, Chapman & Hall, 1996).
- [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] 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] 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] 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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