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## Discussiones Mathematicae Graph Theory

2010 | 30 | 2 | 245-256
Tytuł artykułu

### Chvátal-Erdös type theorems

Autorzy
Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
The Chvátal-Erdös theorems imply that if G is a graph of order n ≥ 3 with κ(G) ≥ α(G), then G is hamiltonian, and if κ(G) > α(G), then G is hamiltonian-connected. We generalize these results by replacing the connectivity and independence number conditions with a weaker minimum degree and independence number condition in the presence of sufficient connectivity. More specifically, it is noted that if G is a graph of order n and k ≥ 2 is a positive integer such that κ(G) ≥ k, δ(G) > (n+k²-k)/(k+1), and δ(G) ≥ α(G)+k-2, then G is hamiltonian. It is shown that if G is a graph of order n and k ≥ 3 is a positive integer such that κ(G) ≥ 4k²+1, δ(G) > (n+k²-2k)/k, and δ(G) ≥ α(G)+k-2, then G is hamiltonian-connected. This result supports the conjecture that if G is a graph of order n and k ≥ 3 is a positive integer such that κ(G) ≥ k, δ(G) > (n+k²-2k)/k, and δ(G) ≥ α(G)+k-2, then G is hamiltonian-connected, and the conjecture is verified for k = 3 and 4.
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EN
Kategorie tematyczne
Wydawca
Czasopismo
Rocznik
Tom
Numer
Strony
245-256
Opis fizyczny
Daty
wydano
2010
otrzymano
2009-01-20
poprawiono
2009-06-25
zaakceptowano
2009-06-25
Twórcy
autor
• University of Alaska at Fairbanks, Fairbanks, AK 99775-6660, USA
autor
• University of Memphis, Memphis, TN 38152, USA
autor
• Emory University, Atlanta, GA 30322, USA
autor
• University of Colorado Denver, Denver, CO 80217, USA
autor
• Lehigh University, Bethlehem, PA 18015, USA
Bibliografia
• [1] G. Chartrand and L. Lesniak, Graphs and Digraphs (Chapman and Hall, London, 1996).
• [2] V. Chvátal and P. Erdös, A note on Hamiltonian circuits, Discrete Math 2 (1972) 111-113, doi: 10.1016/0012-365X(72)90079-9.
• [3] G.A. Dirac, Some theorems on abstract graphs, Proc. London Math. Soc. 2 (1952) 69-81, doi: 10.1112/plms/s3-2.1.69.
• [4] H. Enomoto, Long paths and large cycles in finite graphs, J. Graph Theory 8 (1984) 287-301, doi: 10.1002/jgt.3190080209.
• [5] P. Fraisse, $D_λ$-cycles and their applications for hamiltonian cycles, Thése de Doctorat d'état (Université de Paris-Sud, 1986).
• [6] K. Ota, Cycles through prescribed vertices with large degree sum, Discrete Math. 145 (1995) 201-210, doi: 10.1016/0012-365X(94)00036-I.
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