Saved in:
| Main Authors: | , |
|---|---|
| Format: | Preprint |
| Published: |
2024
|
| Subjects: | |
| Online Access: | https://arxiv.org/abs/2409.07366 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866912022826844160 |
|---|---|
| author | Lima, Paloma T. Nikabadi, Amir |
| author_facet | Lima, Paloma T. Nikabadi, Amir |
| contents | A family $\mathcal{F}$ of graphs is a \textit{Gallai family} if for every connected graph $G\in \mathcal{F}$, all longest paths in $G$ have a common vertex. While it is not known whether $P_5$-free graphs are a Gallai family, Long Jr., Milans, and Munaro [The Electronic Journal of Combinatorics, 2023] showed that this is \emph{not} the case for the class of claw-free graphs. We give a complete characterization of the graphs $H$ of size at most five for which $(\text{claw}, H)$-free graphs form a Gallai family. We also show that $(P_5, H)$-free graphs form a Gallai family if $H$ is a triangle, a paw, or a diamond. Both of our results are constructive. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_07366 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Non-empty intersection of longest paths in $P_5$-free and claw-free graphs Lima, Paloma T. Nikabadi, Amir Combinatorics A family $\mathcal{F}$ of graphs is a \textit{Gallai family} if for every connected graph $G\in \mathcal{F}$, all longest paths in $G$ have a common vertex. While it is not known whether $P_5$-free graphs are a Gallai family, Long Jr., Milans, and Munaro [The Electronic Journal of Combinatorics, 2023] showed that this is \emph{not} the case for the class of claw-free graphs. We give a complete characterization of the graphs $H$ of size at most five for which $(\text{claw}, H)$-free graphs form a Gallai family. We also show that $(P_5, H)$-free graphs form a Gallai family if $H$ is a triangle, a paw, or a diamond. Both of our results are constructive. |
| title | Non-empty intersection of longest paths in $P_5$-free and claw-free graphs |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2409.07366 |