# DMGT

ISSN 1234-3099 (print version)

ISSN 2083-5892 (electronic version)

# IMPACT FACTOR 2018: 0.741

SCImago Journal Rank (SJR) 2018: 0.763

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# Discussiones Mathematicae Graph Theory

## 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

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