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The longest cycle of a graph with a large minimal degree

We show that every graph G on n vertices with minimal degree at least n/k contains a cycle of length at least [n/(k − 1)]. This verifies a conjecture of Katchalski. When k = 2 our result reduces to the classical theorem of Dirac that asserts that if all degrees are at least 1/2n then G is Hamiltonian.