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Discussiones Mathematicae Graph Theory 25(3) (2005) 273-289
DOI: https://doi.org/10.7151/dmgt.1281

POTENTIAL FORBIDDEN TRIPLES IMPLYING HAMILTONICITY:
FOR SUFFICIENTLY LARGE GRAPHS

Ralph J. Faudree University of Memphis, Memphis, TN 38152, USA

Ronald J. Gould Emory University, Atlanta, GA 30322, USA

Michael S. Jacobson University of Colorado at Denver Denver, CO 80217, USA

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.

Keywords: hamiltonian, forbidden subgraph, claw-free, induced subgraph.

2000 Mathematics Subject Classification: 05C45.

References

[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.
[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.
[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.
[7]	R.J. Faudree, R.J. Gould and M.S. Jacobson, Forbidden triples including K1,3 implying hamiltonicity: for sufficiently large graphs, preprint.
[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.

Received 20 December 2003
Revised 14 June 2005