New Sufficient Conditions for s-Hamiltonian Graphs and s-Hamiltonian Connected Graphs

A graph G is s-Hamiltonian if for any S ⊆ V (G) of order at most s, G−S has a Hamiltonian-cycle, and s-Hamiltonian connected if for any S ⊆ V (G) of order at most s, G − S is Hamiltonian-connected. Let k > 0, s ≥ 0 be two integers. The following are proved in this paper: (1) Let k ≥ s+ 2 and s ≤ n− 3. If G is a k-connected graph of order n and if max{d(v) : v ∈ I} ≥ (n+s)/2 for every independent set I of order k−s such that I has two distinct vertices x, y with 1 ≤ |N(x)∩N(y)| ≤ α(G)+ s−1, then G is s-Hamiltonian. (2) Let k ≥ s + 3 and s ≤ n − 2. If G is a k-connected graph of order n and if max{d(v) : v ∈ I} ≥ (n + s + 1)/2 for every independent set I of order k − s− 1 such that I has two distinct vertices x, y with 1 ≤ |N(x) ∩N(y)| ≤ α(G) + s, then G is s-Hamiltonian connected. These extended several former results by Dirac, Ore, Fan and Chen.