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.
@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/
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