Towards a conjecture on long induced rainbow paths in triangle-free graphs
Fuente:
arXiv
Saved in:
| Main Authors: | , , |
|---|---|
| Format: | Preprint |
| Published: |
2026
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866914230269116416 |
|---|---|
| author | Aravind, N. R. Gupta, Shiwali Mathew, Rogers |
| author_facet | Aravind, N. R. Gupta, Shiwali Mathew, Rogers |
| contents | Given a triangle-free graph $G$ with chromatic number $k$ and a proper vertex coloring $ϕ$ of $G$, it is conjectured that $G$ contains an induced rainbow path on $k$ vertices under $ϕ$. Scott and Seymour proved the existence of an induced rainbow path on $(\log \log \log k)^{\frac{1}{3}- o(1)}$ vertices. We improve this to $(\log k)^{\frac{1}{2}- o(1)}$ vertices. Further, we prove the existence of an induced path that sees $\frac{k}{2}$ colors. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_00602 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Towards a conjecture on long induced rainbow paths in triangle-free graphs Aravind, N. R. Gupta, Shiwali Mathew, Rogers Combinatorics Discrete Mathematics 05C15, 05C35, 05C38, 05D10 G.2.2 Given a triangle-free graph $G$ with chromatic number $k$ and a proper vertex coloring $ϕ$ of $G$, it is conjectured that $G$ contains an induced rainbow path on $k$ vertices under $ϕ$. Scott and Seymour proved the existence of an induced rainbow path on $(\log \log \log k)^{\frac{1}{3}- o(1)}$ vertices. We improve this to $(\log k)^{\frac{1}{2}- o(1)}$ vertices. Further, we prove the existence of an induced path that sees $\frac{k}{2}$ colors. |
| title | Towards a conjecture on long induced rainbow paths in triangle-free graphs |
| topic | Combinatorics Discrete Mathematics 05C15, 05C35, 05C38, 05D10 G.2.2 |
| url | https://arxiv.org/abs/2601.00602 |