Towards a conjecture on degree conditions for Ramsey goodness of paths
Fuente:
arXiv
Guardado en:
| Autor principal: | |
|---|---|
| Formato: | Preprint |
| Publicado: |
2026
|
| Materias: | |
| Acceso en línea: | |
| Etiquetas: |
Agregar Etiqueta
Sin Etiquetas, Sea el primero en etiquetar este registro!
|
| _version_ | 1866908993205567488 |
|---|---|
| author | You, Chunlin |
| author_facet | You, Chunlin |
| contents | Recently, Aragão, Marciano, and Mendonça [\emph{European J. Combin.}, 2025] conjectured that for any graph $G$ on $n$ vertices satisfying $(r-1)(t-1)k < n \le (r-1)(t-1)(k+1)$, the minimum degree condition $δ(G) \ge n - \left\lceil \frac{k}{k+1} \left\lceil \frac{n}{r-1} \right\rceil \right\rceil$ guarantees that $G \rightarrow (K_r, P_t)$. In this paper, we prove their conjecture for the regime $k \ge t-3$. Because the parameter $k$ scales linearly with the host graph order $n$, our result establishes the asymptotic truth of the conjecture. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_23131 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Towards a conjecture on degree conditions for Ramsey goodness of paths You, Chunlin Combinatorics 05D10, 05C55 Recently, Aragão, Marciano, and Mendonça [\emph{European J. Combin.}, 2025] conjectured that for any graph $G$ on $n$ vertices satisfying $(r-1)(t-1)k < n \le (r-1)(t-1)(k+1)$, the minimum degree condition $δ(G) \ge n - \left\lceil \frac{k}{k+1} \left\lceil \frac{n}{r-1} \right\rceil \right\rceil$ guarantees that $G \rightarrow (K_r, P_t)$. In this paper, we prove their conjecture for the regime $k \ge t-3$. Because the parameter $k$ scales linearly with the host graph order $n$, our result establishes the asymptotic truth of the conjecture. |
| title | Towards a conjecture on degree conditions for Ramsey goodness of paths |
| topic | Combinatorics 05D10, 05C55 |
| url | https://arxiv.org/abs/2604.23131 |