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.

- Hamiltonian,
- Hamiltonian-connected,
- Chvátal-Erdös condition,
- Independence number

*Discussiones Mathematicae Graph Theory*30.2 (2010): 245-256.

doi:10.7151/dmgt.1490

Available at: http://works.bepress.com/colton_magnant/6

Discussiones Mathematicae Graph Theory.