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arXiv · 2508.07997

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

Maximilian Krone
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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 $k+k^{\frac{1}{2}+\alpha} = \big(1+o(1)\big)k$ is sufficient for a $(k+1)$-edge-connected subgraph. It is shown that all of these results are sharp in some sense. The results are applied to decompose graphs into two highly connected or edge-connected parts.

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Maximilian Krone. 2025-08-11. Sufficient minimum degree conditions for the existence of highly connected or edge-connected subgraphs. https://arxiv.org/abs/2508.07997

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