Abstract
A graph G with at least three vertices is said to be Hamiltonian-connected if each pair distinct vertices is connected by a Hamiltonian path. If uv is a edge of Hamiltonian-connected graph G, then there must exist a Hamiltonian cycle containing uv. If C1, C2,...,Cr are distinct Hamiltonian cycles, and if edge xy ∉E(c1∪ c2 ∪...cr), then we can obtain Hamiltonian cycle Cr+1 containing xy and clearly Cr+1 is different from, C1 C2 Cr. Thus, a Hamiltonianconnected graph has many Hamiltonian cycles, so we can see that the sufficient conditions of Hamiltonian-connected is stronger than Hamiltonian and pancyclic. In 1989 Faudree et al., considered 3-connected graphs and proved that if G is a 3-connected graph of order n and NC ≥ 2n/3, then G is Hamiltonian-connected graph. In this present paper we investigate some further sufficient conditions of Hamiltonian-connected graphs.
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