Sufficient minimum degree conditions for the existence of highly connected or edge-connected subgraphs
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arXiv
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| Natura: | Preprint |
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2025
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| _version_ | 1866911727657943040 |
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| author | Krone, Maximilian |
| author_facet | Krone, Maximilian |
| contents | 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 $α>0$ and for $k$ large enough in terms of $α$, already a minimum degree of at least $k+k^{\frac{1}{2}+α} = \big(1+o(1)\big)k$ 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_07997 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Sufficient minimum degree conditions for the existence of highly connected or edge-connected subgraphs Krone, Maximilian Combinatorics 05C40, 05C35 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 $α>0$ and for $k$ large enough in terms of $α$, already a minimum degree of at least $k+k^{\frac{1}{2}+α} = \big(1+o(1)\big)k$ 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. |
| title | Sufficient minimum degree conditions for the existence of highly connected or edge-connected subgraphs |
| topic | Combinatorics 05C40, 05C35 |
| url | https://arxiv.org/abs/2508.07997 |