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| Main Author: | |
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| Format: | Preprint |
| Published: |
2023
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2312.11095 |
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| _version_ | 1866914955002904576 |
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| author | Wolf, Isaak H. |
| author_facet | Wolf, Isaak H. |
| contents | Let $G$ be a simple graph and let $n,m$ be two integers with $0<m<n$. We prove that $iso(G-S)\leq \frac{n}{m}|S|$ for every $S \subset V(G)$ if and only if $G$ has a $\{C_{2i+1},T \colon 1 \leq i < \frac{m}{n-m}, T\in\mathcal{T}_{\frac{n}{m}}\}$-factor, where $iso(G-S)$ denotes the number of isolated vertices of $G-S$ and $\mathcal{T}_{\frac{n}{m}}$ is a special family of trees. Furthermore, we characterize the trees in $\mathcal{T}_{\frac{n}{m}}$ in terms of their bipartition. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2312_11095 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Fractional factors and component factors in graphs with isolated toughness smaller than 1 Wolf, Isaak H. Combinatorics Let $G$ be a simple graph and let $n,m$ be two integers with $0<m<n$. We prove that $iso(G-S)\leq \frac{n}{m}|S|$ for every $S \subset V(G)$ if and only if $G$ has a $\{C_{2i+1},T \colon 1 \leq i < \frac{m}{n-m}, T\in\mathcal{T}_{\frac{n}{m}}\}$-factor, where $iso(G-S)$ denotes the number of isolated vertices of $G-S$ and $\mathcal{T}_{\frac{n}{m}}$ is a special family of trees. Furthermore, we characterize the trees in $\mathcal{T}_{\frac{n}{m}}$ in terms of their bipartition. |
| title | Fractional factors and component factors in graphs with isolated toughness smaller than 1 |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2312.11095 |