Chvátal-Erdös type theorems
Jill R. Faudree ; Ralph J. Faudree ; Ronald J. Gould ; Michael S. Jacobson ; Colton Magnant
Discussiones Mathematicae Graph Theory, Tome 30 (2010), p. 245-256 / Harvested from The Polish Digital Mathematics Library

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.

Publié le : 2010-01-01
EUDML-ID : urn:eudml:doc:270915
@article{bwmeta1.element.bwnjournal-article-doi-10_7151_dmgt_1490,
     author = {Jill R. Faudree and Ralph J. Faudree and Ronald J. Gould and Michael S. Jacobson and Colton Magnant},
     title = {Chv\'atal-Erd\"os type theorems},
     journal = {Discussiones Mathematicae Graph Theory},
     volume = {30},
     year = {2010},
     pages = {245-256},
     zbl = {1214.05069},
     language = {en},
     url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_7151_dmgt_1490}
}
Jill R. Faudree; Ralph J. Faudree; Ronald J. Gould; Michael S. Jacobson; Colton Magnant. Chvátal-Erdös type theorems. Discussiones Mathematicae Graph Theory, Tome 30 (2010) pp. 245-256. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_7151_dmgt_1490/

[000] [1] G. Chartrand and L. Lesniak, Graphs and Digraphs (Chapman and Hall, London, 1996). | Zbl 0890.05001

[001] [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. | Zbl 0233.05123

[002] [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. | Zbl 0047.17001

[003] [4] H. Enomoto, Long paths and large cycles in finite graphs, J. Graph Theory 8 (1984) 287-301, doi: 10.1002/jgt.3190080209. | Zbl 0544.05044

[004] [5] P. Fraisse, Dλ-cycles and their applications for hamiltonian cycles, Thése de Doctorat d’état (Université de Paris-Sud, 1986).

[005] [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. | Zbl 0838.05071