The Hamiltonian index of graphs

The Hamiltonian index of a graph G is defined as h ( G ) = min { m : L m ( 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 H ̃ ( m ) ( G ) from G and prove that if h ( G ) ≥ 2 , then h ( G ) = min { m : H ̃ ( m ) ( G )  has a spanning Eulerian subgraph } .