Andr{\'a}sfai--Erd\H{o}s--S\'{o}s theorem under max-degree constraints

We establish the following strengthening of the celebrated Andr{\'a}sfai--Erd\H{o}s--S\'{o}s theorem: If $G$ is an $n$-vertex $K_{r+1}$-free graph whose minimum degree $\delta(G)$ and maximum degree $\Delta(G)$ satisfy \begin{align*} \delta(G) > \min \left\{ \frac{3r-4}{3r-2}n-\frac{\Delta(G)}{3r-2},~n-\frac{\Delta(G)+1}{r-1} \right\}, \end{align*} then $G$ is $r$-partite. This bound is tight for all feasible values of $\Delta(G)$. We also obtain an analogous tight result for graphs with large odd girth. Our proof does not rely on the Andr{\'a}sfai--Erd\H{o}s--S\'{o}s theorem itself, and therefore yields an alternative proof of this classical result.