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 ⌊ 2 t + 2 3 ⌋ for every nonnegative integer t . The upper bound is sharp.