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

TL;DR

This paper establishes minimum degree conditions that guarantee the existence of highly connected or edge-connected subgraphs, extending classical conjectures and providing sharp bounds for various graph classes.

Contribution

It proves new minimum degree thresholds for the existence of large, highly connected or edge-connected subgraphs, including sharp bounds and special cases like triangle-free graphs.

Findings

Graphs with minimum degree at least 3k-1 contain (k+1)-connected subgraphs with more than 2k vertices.

Triangle-free graphs with average degree at least 2k have (k+1)-connected subgraphs with at least 2(k+1) vertices.

Graphs with average degree at least 2k contain (k+1)-edge-connected subgraphs on more than 2k vertices.

Abstract

Mader (1979) conjectured 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 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 $\alpha>0$ and for $k$ large enough in terms of $\alpha$, already a minimum degree of at least $k+k^{\frac{1}{2}+\alpha} = \big(1+o(1)\big)k$ is sufficient for a $(k+1)$-edge-connected subgraph. It is shown that all of these results are sharp in some sense. The results may be applied to decompose graphs into two highly connected or edge-connected parts.