Andr{á}sfai--Erdős--Sós theorem under max-degree constraints

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 $\delta (G)$ and maximum degree $\mathrm{\Delta }(G)$ 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 $\mathrm{\Delta }(G)$. 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.