Andr{á}sfai--Erdős--Sós theorem under max-degree constraints
Fuente:
arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866911312765779968 |
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| author | Liu, Xizhi Ren, Sijie Wang, Jian |
| author_facet | Liu, Xizhi Ren, Sijie Wang, Jian |
| contents | We establish the following strengthening of the celebrated Andr{á}sfai--Erdős--Sós theorem: If $G$ is an $n$-vertex $K_{r+1}$-free graph whose minimum degree $δ(G)$ and maximum degree $Δ(G)$ satisfy
\begin{align*}
δ(G) > \min \left\{ \frac{3r-4}{3r-2}n-\frac{Δ(G)}{3r-2},~n-\frac{Δ(G)+1}{r-1} \right\},
\end{align*}
then $G$ is $r$-partite. This bound is tight for all feasible values of $Δ(G)$. We also obtain an analogous tight result for graphs with large odd girth.
Our proof does not rely on the Andr{á}sfai--Erdős--Sós theorem itself, and therefore yields an alternative proof of this classical result. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_10190 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Andr{á}sfai--Erdős--Sós theorem under max-degree constraints Liu, Xizhi Ren, Sijie Wang, Jian Combinatorics We establish the following strengthening of the celebrated Andr{á}sfai--Erdős--Sós theorem: If $G$ is an $n$-vertex $K_{r+1}$-free graph whose minimum degree $δ(G)$ and maximum degree $Δ(G)$ satisfy \begin{align*} δ(G) > \min \left\{ \frac{3r-4}{3r-2}n-\frac{Δ(G)}{3r-2},~n-\frac{Δ(G)+1}{r-1} \right\}, \end{align*} then $G$ is $r$-partite. This bound is tight for all feasible values of $Δ(G)$. We also obtain an analogous tight result for graphs with large odd girth. Our proof does not rely on the Andr{á}sfai--Erdős--Sós theorem itself, and therefore yields an alternative proof of this classical result. |
| title | Andr{á}sfai--Erdős--Sós theorem under max-degree constraints |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2512.10190 |