The hamiltonian index of a 2-connected graph

Let GG be a graph. Then the hamiltonian index h(G)h(G) of GG is the smallest number of iterations of line graph operator that yield a hamiltonian graph. In this paper we show that h(G)≤max{1,|V(G)|−Δ(G)3} for every 2-connected simple graph GG that is not isomorphic to the graph obtained from a dipole with three parallel edges by replacing every edge by a path of length l≥3l≥3. We also show that max{h(G),h(G¯)}≤|V(G)|−36 for any two 2-connected nonhamiltonian graphs GG and G¯ with at least 74 vertices. The upper bounds are all sharp.