The hamiltonian index of a 2-connected graph

Abstract

Let G be a graph. Then the hamiltonian index h ( G ) of G 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 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 l ≥ 3 . We also show that max { h ( G ) , h ( G ¯ ) } ≤ | V ( G ) | − 3 6 for any two 2-connected nonhamiltonian graphs G and G ¯ with at least 74 vertices. The upper bounds are all sharp.