The Chvátal-Erdös condition for supereulerian graphs and the Hamiltonian index

A classical result of Chvátal and Erdös says that the graph G with connectivity κ (G) not less than its independence number α (G) (i.e. κ (G) ≥ α (G)) is Hamiltonian. In this paper, we show that the 2-connected graph G with κ (G) ≥ α (G) - 1 is one of the following: supereulerian, the Petersen graph, the 2-connected graph with three vertices of degree two obtained from K_{2, 3} by replacing a vertex of degree three with a triangle, one of the 2-connected graphs obtained from K_{2, 3} 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 K_{2, 3} by replacing at most two branches of K_{2, 3} with a branch of length three. We also show that the Hamiltonian index of the simple 2-connected graph G with κ (G) ≥ α (G) - t is at most ⌊ frac(2 t + 2, 3) ⌋ for every nonnegative integer t. The upper bound is sharp.