Spanning tree packing and 2-essential edge-connectivity

Abstract

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