Trees and near-linear stable sets
Fuente:
arXiv
Enregistré dans:
| Auteurs principaux: | , , |
|---|---|
| Format: | Preprint |
| Publié: |
2024
|
| Sujets: | |
| Accès en ligne: | |
| Tags: |
Ajouter un tag
Pas de tags, Soyez le premier à ajouter un tag!
|
| _version_ | 1866908455113064448 |
|---|---|
| author | Nguyen, Tung Scott, Alex Seymour, Paul |
| author_facet | Nguyen, Tung Scott, Alex Seymour, Paul |
| contents | When $H$ is a forest, the Gyárfás-Sumner conjecture implies that every graph $G$ with no induced subgraph isomorphic to $H$ and with bounded clique number has a stable set of linear size. We cannot prove that, but we prove that every such graph $G$ has a stable set of size $|G|^{1-o(1)}$. If $H$ is not a forest, there need not be such a stable set.
Second, we prove that when $H$ is a ``multibroom'', there {\em is} a stable set of linear size. As a consequence, we deduce that all multibrooms satisfy a ``fractional colouring'' version of the Gyárfás-Sumner conjecture.
Finally, we discuss extensions of our results to the multicolour setting. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_09397 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Trees and near-linear stable sets Nguyen, Tung Scott, Alex Seymour, Paul Combinatorics When $H$ is a forest, the Gyárfás-Sumner conjecture implies that every graph $G$ with no induced subgraph isomorphic to $H$ and with bounded clique number has a stable set of linear size. We cannot prove that, but we prove that every such graph $G$ has a stable set of size $|G|^{1-o(1)}$. If $H$ is not a forest, there need not be such a stable set. Second, we prove that when $H$ is a ``multibroom'', there {\em is} a stable set of linear size. As a consequence, we deduce that all multibrooms satisfy a ``fractional colouring'' version of the Gyárfás-Sumner conjecture. Finally, we discuss extensions of our results to the multicolour setting. |
| title | Trees and near-linear stable sets |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2409.09397 |