Non-empty intersection of longest paths in $H$-free graphs
Fuente:
arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2023
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| _version_ | 1866913801430892544 |
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| author | Long Jr., James A. Milans, Kevin G. Munaro, Andrea |
| author_facet | Long Jr., James A. Milans, Kevin G. Munaro, Andrea |
| contents | We make progress toward a characterization of the graphs $H$ such that every connected $H$-free graph has a longest path transversal of size $1$. In particular, we show that the graphs $H$ on at most $4$ vertices satisfying this property are exactly the linear forests. We also show that if the order of a connected graph $G$ is large relative to its connectivity $κ(G)$, and its independence number $α(G)$ satisfies $α(G) \le κ(G) + 2$, then each vertex of maximum degree forms a longest path transversal of size $1$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2302_07110 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Non-empty intersection of longest paths in $H$-free graphs Long Jr., James A. Milans, Kevin G. Munaro, Andrea Combinatorics Discrete Mathematics We make progress toward a characterization of the graphs $H$ such that every connected $H$-free graph has a longest path transversal of size $1$. In particular, we show that the graphs $H$ on at most $4$ vertices satisfying this property are exactly the linear forests. We also show that if the order of a connected graph $G$ is large relative to its connectivity $κ(G)$, and its independence number $α(G)$ satisfies $α(G) \le κ(G) + 2$, then each vertex of maximum degree forms a longest path transversal of size $1$. |
| title | Non-empty intersection of longest paths in $H$-free graphs |
| topic | Combinatorics Discrete Mathematics |
| url | https://arxiv.org/abs/2302.07110 |