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Discussiones Mathematicae Graph Theory

2007 | 27 | 2 | 229-240
Tytuł artykułu

Variations on a sufficient condition for Hamiltonian graphs

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Treść / Zawartość
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Języki publikacji
EN
Abstrakty
EN
Given a 2-connected graph G on n vertices, let G* be its partially square graph, obtained by adding edges uv whenever the vertices u,v have a common neighbor x satisfying the condition $N_G(x) ⊆ N_G[u] ∪ N_G[v]$, where $N_G[x] = N_G(x) ∪ {x}$. In particular, this condition is satisfied if x does not center a claw (an induced $K_{1,3}$). Clearly G ⊆ G* ⊆ G², where G² is the square of G. For any independent triple X = {x,y,z} we define
σ̅(X) = d(x) + d(y) + d(z) - |N(x) ∩ N(y) ∩ N(z)|.
Flandrin et al. proved that a 2-connected graph G is hamiltonian if [σ̅]₃(X) ≥ n holds for any independent triple X in G. Replacing X in G by X in the larger graph G*, Wu et al. improved recently this result. In this paper we characterize the nonhamiltonian 2-connected graphs G satisfying the condition [σ̅]₃(X) ≥ n-1 where X is independent in G*. Using the concept of dual closure we (i) give a short proof of the above results and (ii) we show that each graph G satisfying this condition is hamiltonian if and only if its dual closure does not belong to two well defined exceptional classes of graphs. This implies that it takes a polynomial time to check the nonhamiltonicity or the hamiltonicity of such G.
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EN
Kategorie tematyczne
Wydawca
Czasopismo
Rocznik
Tom
Numer
Strony
229-240
Opis fizyczny
Daty
wydano
2007
otrzymano
2005-09-23
poprawiono
2007-03-12
zaakceptowano
2007-03-12
Twórcy
autor
• UAG - CEREGMIA - GRIMAAG, B.P. 7209, 97275 Schoelcher Cedex, Martinique, France
autor
• UAG - CEREGMIA - GRIMAAG, 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, An improvement of Fraisse's sufficient condition for hamiltonian graphs, J. Graph Theory 16 (1992) 529-543, doi: 10.1002/jgt.3190160602.
• [3] A. Ainouche, O. Favaron and H. Li, Global insertion and hamiltonicity in DCT-graphs, Discrete Math. 184 (1998) 1-13, doi: 10.1016/S0012-365X(97)00157-X.
• [4] A. Ainouche and M. Kouider, Hamiltonism and Partially Square Graphs, Graphs and Combinatorics 15 (1999) 257-265, doi: 10.1007/s003730050059.
• [5] A. Ainouche and I. Schiermeyer, 0-dual closures for several classes of graphs, Graphs and Combinatorics 19 (2003) 297-307, doi: 10.1007/s00373-002-0523-y.
• [6] A. Ainouche, Extension of several sufficient conditions for hamiltonian graphs, Discuss. Math. Graph Theory 26 (2006) 23-39, doi: 10.7151/dmgt.1298.
• [7] J.A. Bondy and U.S.R. Murty, Graph Theory with Applications (Macmillan, London, 1976.)
• [8] J.A. Bondy and V. Chvátal, A method in graph theory, Discrete Math. 15 (1976) 111-135, doi: 10.1016/0012-365X(76)90078-9.
• [9] 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.
• [10] Z. Wu, X. Zhang and X. Zhou, Hamiltonicity, neighborhood intersections and the partially square graphs, Discrete Math. 242 (2002) 245-254, doi: 10.1016/S0012-365X(00)00394-0.
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Bibliografia
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