Paper
Andr{á}sfai--Erdős--Sós theorem under max-degree constraints
Authors
Xizhi Liu, Sijie Ren, Jian Wang
Abstract
We establish the following strengthening of the celebrated Andr{á}sfai--Erdős--Sós theorem: If is an -vertex -free graph whose minimum degree and maximum degree 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 is -partite. This bound is tight for all feasible values of . 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.