Extension of several sufficient conditions for Hamiltonian graphs

• Autorzy: Ahmed Ainouche
• Języki publikacji: EN

Abstrakty

EN

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 $(K_r ∪ Kₛ ∪ Kₜ) ∨ K₂$, 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.

Słowa kluczowe

EN

hamiltonian graph; dual closure; neighborhood closure

Kategorie tematyczne

• 05C45: Eulerian and Hamiltonian graphs
• 05C38: Paths and cycles

• Wydawca: De Gruyter Open
• Czasopismo: Discussiones Mathematicae Graph Theory
• Rocznik: 2006
• Tom: 26
• Numer: 1
• Strony: 23-39

Daty

wydano

2006

otrzymano

2004-09-21

poprawiono

2005-09-22

Twórcy

autor

Ahmed Ainouche

• CEREGMIA-GRIMAAG, Campus de Schoelcher, B.P. 7209, 97275 Schoelcher Cedex, Martinique, France

Bibliografia

• [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.

Identyfikatory

DOI

10.7151/dmgt.1298