PDF

Discussiones Mathematicae Graph Theory 26(1) (2006) 23-39
DOI: https://doi.org/10.7151/dmgt.1298

EXTENSION OF SEVERAL SUFFICIENT CONDITIONS FOR HAMILTONIAN GRAPHS

Ahmed Ainouche

CEREGMIA-GRIMAAG
Campus de Schoelcher
B.P. 7209
97275 Schoelcher Cedex
Martinique, France
e-mail: a.ainouche@martinique.univ-ag.fr

Abstract

Let G be a 2-connected graph of order n. Suppose that for all 3-independent sets X in G, there exists a vertex u in X such that |N(X∖{u})|+d(u) ≥ n−1. Using the concept of dual closure, we prove that

1. G is hamiltonian if and only if its 0-dual closure is either complete or the cycle C7

2. G is nonhamiltonian if and only if its 0-dual closure is either the graph (Kr∪Ks ∪Kt)∨ K2, 1 ≤ r ≤ s ≤ t or the graph ([(n+1)/2])K1∨ K[(n−1)/2].

It follows that it takes a polynomial time to check the hamiltonicity or the nonhamiltonicity of a graph satisfying the above condition. From this main result we derive a large number of extensions of previous sufficient conditions for hamiltonian graphs. All these results are sharp.

Keywords: hamiltonian graph, dual closure, neighborhood closure.

2000 Mathematics Subject Classification: 05C38, 05C45.

References

[1]	A. Ainouche and N. Christofides, Semi-independence number of a graph and the existence of hamiltonian circuits, Discrete Applied Math. 17 (1987) 213-221, doi: 10.1016/0166-218X(87)90025-4.
[2]	A. Ainouche, A common generalization of Chvàtal-Erdös and Fraisse's sufficient conditions for hamiltonian graphs, Discrete Math. 142 (1995) 1-19, doi: 10.1016/0012-365X(94)00002-Z.
[3]	A. Ainouche, Extensions of a closure condition: the β-closure (Rapport de Recherche CEREGMIA, 2001).
[4]	A. Ainouche and I. Schiermeyer, 0-dual closure for several classes of graphs, Graphs and Combinatorics 19 (2003) 297-307, doi: 10.1007/s00373-002-0523-y.
[5]	A. Ainouche and S. Lapiquonne, Variations on a strong sufficient condition for hamiltonian graphs, submitted.
[6]	J.A. Bondy, Longest paths and cycles in graphs of high degree, Research Report CORR 80-16, Dept. of Combinatorics and Optimization, University of Waterloo, Ont. Canada.
[7]	J.A. Bondy and V. Chvàtal, A method in graph theory, Discrete Math. 15 (1976) 111-135, doi: 10.1016/0012-365X(76)90078-9.
[8]	G. Chen, One sufficient condition for Hamiltonian Graphs, J. Graph Theory 14 (1990) 501-508, doi: 10.1002/jgt.3190140414.
[9]	G.A. Dirac, Some theorems on abstract graphs, Proc. London Math. Soc. 2 (1952) 69-81, doi: 10.1112/plms/s3-2.1.69.
[10]	R.J. Faudree, R.J. Gould, M.S. Jacobson and L.S. Lesniak, Neighborhood unions and highly Hamiltonian Graphs, Ars Combin. 31 (1991) 139-148.
[11]	R.J. Faudree, R.J. Gould, M.S. Jacobson and R.H. Shelp, Neighborhood unions and Hamiltonian properties in Graphs, J. Combin. Theory (B) 47 (1989) 1-9, doi: 10.1016/0095-8956(89)90060-9.
[12]	E. Flandrin, H.A. Jung and H. Li, Hamiltonism, degrees sums and neighborhood intersections, Discrete Math. 90 (1991) 41-52, doi: 10.1016/0012-365X(91)90094-I.
[13]	P. Fraisse, A new sufficient condition for Hamiltonian Graphs, J. Graph Theory 10 (1986) 405-409, doi: 10.1002/jgt.3190100316.
[14]	T. Feng, A note on the paper A new sufficient condition for hamiltonian graph, Systems Sci. Math. Sci. 5 (1992) 81-83.
[15]	I. Schiermeyer, Computation of the 0-dual closure for hamiltonian graphs, Discrete Math. 111 (1993) 455-464, doi: 10.1016/0012-365X(93)90183-T.
[16]	O. Ore, Note on Hamiltonian circuits, Am. Math. Monthly 67 (1960) 55, doi: 10.2307/2308928.

Received 21 September 2004
Revised 22 September 2005