A classical result of Chvátal and Erdös says that the graph with connectivity not less than its independence number (i.e. ) is Hamiltonian. In this paper, we show that the 2-connected graph with is one of the following: supereulerian, the Petersen graph, the 2-connected graph with three vertices of degree two obtained from by replacing a vertex of degree three with a triangle, one of the 2-connected graphs obtained from by replacing a vertex of degree two with a complete graph of order at least three and by replacing at most one branch of length two in the resulting graph with a branch of length three, or one of the graphs obtained from by replacing at most two branches of with a branch of length three. We also show that the Hamiltonian index of the simple 2-connected graph with is at most for every nonnegative integer . The upper bound is sharp.