Sufficient minimum degree conditions for the existence of highly connected or edge-connected subgraphs

Mader conjectured that an average degree of at least 3k-1 is sufficient for the existence of a (k+1) -connected subgraph. The following minimum degree version holds: Every graph with minimum degree at least 3k-1 has a (k+1) -connected subgraph on more than 2k vertices. Moreover, for triangle-free graphs, already an average degree of at least 2k is sufficient for a (k+1) -connected subgraph (which has at least 2 (k+1) vertices). For edge-connectedness (in simple graphs), we prove the following: Every graph of average degree at least 2k has a (k+1) -edge-connected subgraph on more than 2k vertices. Moreover, for every small α>0 and for k large enough in terms of α, already a minimum degree of at least k+k^1{2+α} = (1+o (1) ) k is sufficient for a (k+1) -edge-connected subgraph. It is shown that all of these results are sharp in some sense. The results can be used for the decomposition of graphs into two highly (edge-) connected parts.