The Hamiltonian index of graphs

The Hamiltonian index of a graph G is defined as h(G)=min{m:Lm(G) is Hamiltonian}. In this paper, using the reduction method of Catlin [P.A. Catlin, A reduction method to find spanning Eulerian subgraphs, J. Graph Theory 12 (1988) 29-44], we constructed a graph H̃(m)(G) from G and prove that if h(G)≥2, then h(G)=min{m:H̃(m)(G) has a spanning Eulerian subgraph}.