The minimum degree of minimal 2-extendable claw-free graphs
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866916988219031552 |
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| author | Guo, Jing Lu, Fuliang Zhang, Heping |
| author_facet | Guo, Jing Lu, Fuliang Zhang, Heping |
| contents | A connected graph $G$ with a perfect matching is said to be $k$-extendable for integers $k$, $1 \leq k\leq \frac{|V(G)|}{2}-1$, if any matching in $G$ of size $k$ is contained in a perfect matching of $G$. A $k$-extendable graph is minimal if the deletion of any edge results in a graph that is not $k$-extendable. In 1994, Plummer proved that every $k$-extendable claw-free graph has minimum degree at least $2k$. Recently, He et al. showed that every minimal 1-extendable graph has minimum degree 2 or 3. In this paper, we prove that the minimum degree of a minimal 2-extendable claw-free graph is either $4$ or $5$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2510_03554 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The minimum degree of minimal 2-extendable claw-free graphs Guo, Jing Lu, Fuliang Zhang, Heping Combinatorics 05C70, 05C07 A connected graph $G$ with a perfect matching is said to be $k$-extendable for integers $k$, $1 \leq k\leq \frac{|V(G)|}{2}-1$, if any matching in $G$ of size $k$ is contained in a perfect matching of $G$. A $k$-extendable graph is minimal if the deletion of any edge results in a graph that is not $k$-extendable. In 1994, Plummer proved that every $k$-extendable claw-free graph has minimum degree at least $2k$. Recently, He et al. showed that every minimal 1-extendable graph has minimum degree 2 or 3. In this paper, we prove that the minimum degree of a minimal 2-extendable claw-free graph is either $4$ or $5$. |
| title | The minimum degree of minimal 2-extendable claw-free graphs |
| topic | Combinatorics 05C70, 05C07 |
| url | https://arxiv.org/abs/2510.03554 |