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Dirac type condition and Hamiltonian-connected graphs

Abstract In 1952 Dirac introduced the Dirac type condition and proved that if G is a connected graph of order n ≥ 3 such that δ(G) ≥ n/2, then G is Hamiltonian. In this paper we consider Hamiltonian-connectedness, which extends the Hamiltonian graphs and prove that if G is a connected graph of order n ≥ 3 such that δ(G) ≥ (n −1)/2, then G is Hamiltonian-connected or G belongs to five families of well-structured graphs. Thus, the condition and the result generalize the above condition and results of Dirac, respectively.