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

Abstract

Mader conjectured in 1979 that an average degree of at least $3k-1$ in a graph is sufficient for the existence of a $(k+1)$-connected subgraph. The following minimum degree analogue holds: Every graph with minimum degree at least $3k-1$ contains 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-connectivity (in simple graphs), we prove the following: Every graph with average degree at least $2k$ contains a $(k+1)$-edge-connected subgraph on more than $2k$ vertices. Moreover, for every small $\alpha >0$ and for $k$ large enough in terms of $\alpha$, already a minimum degree of at least $k+k^{\frac{1}{2}+\alpha }=(1+o(1))k$ is sufficient for a $(k+1)$-edge-connected subgraph. It is shown that all of these results are sharp in some sense. The results are applied to decompose graphs into two highly connected or edge-connected parts.