The hamiltonian index of a 2-connected graph

Let G be a graph. Then the hamiltonian index h(G) of G is the smallest number of iterations of the line graph operator that yield a himiltonian graph. In this paper we show that h(G0 <= max (1, vertical bar V(G)vertical bar-Delta(G)/3) for every 2-connected simple graph G that is not isomorphic to the graph obtained from a dipole with three parallel edges by replacing every edge by a path of length 1 >= 3. We also show that max{h(G), h((G) over bar)} <= vertical bar V(G)vertical bar-3/6 for any two 2-connected nonhamiltonian graphs G and (G) over bar with at least 74 vertices. The upper bounds are all sharp. (c) 2008 Elsevier B.V. All rights reserved.