Strong Turán Stability

Author:

Tyomkyn Mykhaylo,Uzzell Andrew J.

Abstract

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 $k \geq r \geq 2$.

Publisher

The Electronic Journal of Combinatorics

Subject

Computational Theory and Mathematics,Geometry and Topology,Theoretical Computer Science,Applied Mathematics,Discrete Mathematics and Combinatorics

Cited by 8 articles. 订阅此论文施引文献 订阅此论文施引文献,注册后可以免费订阅5篇论文的施引文献,订阅后可以查看论文全部施引文献

1. Improving the Caro–Wei bound and applications to Turán stability;Discrete Applied Mathematics;2024-12

2. Refinement on Spectral Turán’s Theorem;SIAM Journal on Discrete Mathematics;2023-10-19

3. Counterexamples to Gerbner's conjecture on stability of maximal F‐free graphs;Journal of Graph Theory;2023-10-04

4. A stability theorem for maximal C2k+1 ${C}_{2k+1}$‐free graphs;Journal of Graph Theory;2022-03-25

5. A Note on Stability of Odd Cycles;Advances in Applied Mathematics;2022

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