Spanning tree packing and 2-essential edge-connectivity

An edge (vertex) cut X of G is r-essential if G−X has two components each which at least r edges. A graph r-essentially k-edge-connected (resp. k-connected) it no vertex) cuts size less than k. If r=1, we simply call essential. Recently, Lai and Li proved that every m-edge-connected essentially h-edge-connected contains k edge-disjoint spanning trees, where k,m,h are positive integers such k+1≤m≤2k−1 h≥m2m−k−2. In this paper, show 2-essentially not a K5 or fat-triangle with multiplicity andh≥f(m,k)={2m+k−4+k(2k−1)2m−2k−1,m<k+1+8k+14,m+3k−4+k2m−k,m≥k+1+8k+14. Extending Zhan's result, also prove 3-edge-connected 5-edge-connected 8-edge-connected trees. As an application, gives new sufficient condition for Hamilton-connectedness line graphs. 2012, Kaiser Vrána 5-connected minimum degree 6 Hamilton-connected. We allow graphs to have 5 8-connected