Long cycles in graphs with prescribed toughness and minimum degree Original Research Article

A cycle C of a graph G is a Dλ-cycle if every component of G-V(C) has order less than λ. Using the notion of Dλ-cycles, a number of results are established concerning long cycles in graphs with prescribed toughness and minimum degree. Let G be a t-tough graph on n ⩾ 3 vertices. If δ > n/(t + λ) + λ − 2 for some λ ⩽ t + 1, then G contains a Dλ-cycle. In particular, if δ > n/(t + 1) − 1, then G is hamiltonian, improving a classical result of Dirac for t > 1. If G is nonhamiltonian and δ > n/(t + λ) + λ − 2 for some λ ⩽ t + 1, then G contains a cycle of length at least (t + 1)(δ − λ + 2) + t, partially improving another classical result of Dirac for t > 1.