Edge-connectivity and edge-disjoint spanning trees

Given a graph G , for an integer c ∈ { 2 , … , | V ( G ) | } , define λ c ( G ) = min { | X | : X ⊆ E ( G ) , ω ( G − X ) ≥ c } . For a graph G and for an integer c = 1 , 2 , … , | V ( G ) | − 1 , define, τ c ( G ) = min X ⊆ E ( G )  and  ω ( G − X ) > c | X | ω ( G − X ) − c , where the minimum is taken over all subsets X of E ( G ) such that ω ( G − X ) − c > 0 . In this paper, we establish a relationship between λ c ( G ) and τ c − 1 ( G ) , which gives a characterization of the edge-connectivity of a graph G in terms of the spanning tree packing number of subgraphs of G . The digraph analogue is also obtained. The main results are applied to show that if a graph G is s -hamiltonian, then L ( G ) is also s -hamiltonian, and that if a graph G is s -hamiltonian-connected, then L ( G ) is also s -hamiltonian-connected.