We study maximal $K_{r+1}$-free graphs $G$ of almost extremal size—typically, $e(G)=\operatorname{ex}(n,K_{r+1})-O(n)$. We show that any such graph $G$ must have a large amount of `symmetry': in particular, all but very few vertices of $G$ must have twins. (Two vertices $u$ and $v$ are twins if they have the same neighbourhood.) As a corollary, we obtain a new, short proof of a theorem of Simonovits on the structure of extremal $K_{r+1}$-free graphs of chromatic number at least $k$ for all fixed $k \geq r \geq 2$.
@article{10_37236_4717,
author = {Mykhaylo Tyomkyn and Andrew J. Uzzell},
title = {Strong {Tur\'an} stability},
journal = {The electronic journal of combinatorics},
year = {2015},
volume = {22},
number = {3},
doi = {10.37236/4717},
zbl = {1327.05168},
url = {http://geodesic.mathdoc.fr/articles/10.37236/4717/}
}
TY - JOUR
AU - Mykhaylo Tyomkyn
AU - Andrew J. Uzzell
TI - Strong Turán stability
JO - The electronic journal of combinatorics
PY - 2015
VL - 22
IS - 3
UR - http://geodesic.mathdoc.fr/articles/10.37236/4717/
DO - 10.37236/4717
ID - 10_37236_4717
ER -
%0 Journal Article
%A Mykhaylo Tyomkyn
%A Andrew J. Uzzell
%T Strong Turán stability
%J The electronic journal of combinatorics
%D 2015
%V 22
%N 3
%U http://geodesic.mathdoc.fr/articles/10.37236/4717/
%R 10.37236/4717
%F 10_37236_4717
Mykhaylo Tyomkyn; Andrew J. Uzzell. Strong Turán stability. The electronic journal of combinatorics, Tome 22 (2015) no. 3. doi: 10.37236/4717
