The hamiltonian index of a graph and its branch-bonds

Abstract

Let G be an undirected and loopless finite graph that is not a path. The smallest integer m such that the iterated line graph L m ( G) is hamiltonian is called the hamiltonian index of G, denoted by h( G). A reduction method to determine the hamiltonian index of a graph G with h( G)⩾2 is given here. We use it to establish a sharp lower bound and a sharp upper bound on h( G), respectively, thereby improving some known results of Catlin et al. [J. Graph Theory 14 (1990) 347] and Hong-Jian Lai [Discrete Math. 69 (1988) 43]. Examples show that h( G) may reach all integers between the lower bound and the upper bound. We also propose some questions on the topic.