Edge-connectivity and edge-disjoint spanning trees

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