[Paper Review] Strong Tur\'an stability

Mykhaylo Tyomkyn, Andrew J. Uzzell|arXiv (Cornell University)|Sep 30, 2014

Limits and Structures in Graph Theory7 references3 citations

TL;DR

This paper establishes strong stability results for $K_{r+1}$-free graphs near the Turán threshold, showing that such graphs must have large sets of twin vertices (identical neighborhoods) when edge count is $t_{n,r} - O(n\log n)$. The key contribution is a new proof of Simonovits' theorem on extremal graphs with bounded clique number and high chromatic number, using twin vertex structure to show these graphs are blow-ups of bounded-size graphs.

ABSTRACT

We study the behaviour of $K_{r+1}$-free graphs $G$ of almost extremal size, that is, typically, $e(G)=ex(n,K_{r+1})-O(n)$. We show that such graphs must have a large amount of 'symmetry', in particular that all but very few vertices of $G$ must have twins. As a corollary, we obtain a new, short proof of a theorem of Simonovits on the structure of extremal graphs with $\\omega(G)\\leq r$ and $\\chi(G)\\geq k$ for fixed $k \\geq r \\geq 2$.

Motivation & Objective

• To investigate the structural stability of $K_{r+1}$-free graphs with edge counts close to the Turán number $t_{n,r}$, specifically in the linear sub-regime $e(G) = t_{n,r} - O(n\log n)$.
• To understand the emergence of symmetry in extremal graphs, particularly the presence of twin vertices (vertices with identical neighborhoods), as a structural hallmark near the Turán threshold.
• To provide a new, concise proof of Simonovits' theorem on the simplicity (blow-up structure) of extremal $K_{r+1}$-free graphs with high chromatic number and bounded clique number.
• To determine the exact extremal number of edges in $K_{r+1}$-free graphs with clique number $r$ and chromatic number $k = r+2$, and to characterize the structure of such extremal graphs.

Proposed method

• Analyzing the structure of $K_{r+1}$-saturated graphs (maximal $K_{r+1}$-free graphs) with high edge count, focusing on the emergence of twin vertices.
• Using extremal graph theory and Ramsey-theoretic bounds to derive sharp thresholds for the appearance of twin vertices in $K_{r+1}$-saturated graphs.
• Applying a novel stability argument based on neighborhood symmetry to show that graphs with $e(G) \geq t_{n,r} - cn\log n$ must contain a pair of twin vertices for sufficiently large $n$.
• Proving that such graphs are $r$-colorable under mild edge count conditions, leading to a new proof of Simonovits' result on the simplicity of extremal graphs.
• Leveraging known bounds on Ramsey numbers $R(3,t)$ and triangle-free graphs of high chromatic number to derive asymptotic bounds on $\Lambda_r(k)$, the maximum number of vertices in a $K_{r+1}$-free graph with $\omega(G)=r$ and $\chi(G)\geq k$.
• Using a constructive method to build extremal graphs with $\chi(G) = r+2$ and $\omega(G) = r$, and proving that $\Lambda_r(r+2) = 2$ for $r \geq 3$.

Experimental results

Research questions

• RQ1 What is the minimal edge count $e(G)$ such that every $K_{r+1}$-saturated graph $G$ of order $n$ must contain a pair of twin vertices?
• RQ2 Under what conditions does a $K_{r+1}$-free graph with $e(G) = t_{n,r} - O(n\log n)$ necessarily have a large set of twin vertices?
• RQ3 Can the extremal graphs for $\omega(G) \leq r$ and $\chi(G) \geq k$ be shown to be blow-ups of bounded-size graphs, and what is the exact extremal edge count for $k = r+2$?
• RQ4 How do the thresholds for structural simplicity (blow-up structure) differ between $r=2$ and $r \geq 3$ in $K_{r+1}$-saturated graphs?
• RQ5 What is the exact value of $\Lambda_r(k)$, the maximum number of vertices in a $K_{r+1}$-free graph with $\omega(G) = r$ and $\chi(G) \geq k$, particularly for $k = r+2$?

Key findings

• For every $r \geq 2$, there exists a constant $c > 0$ such that every sufficiently large $(r+1)$-saturated graph $G$ with $e(G) \geq t_{n,r} - cn\log n$ contains at least one pair of twin vertices.
• The threshold for the appearance of twin vertices is $t_{n,r} - O(n\log n)$, which is significantly closer to the Turán number than the $o(n^2)$ threshold in the Erd\'os--Simonovits stability theorem.
• For $r=2$, every $3$-saturated graph with $e(G) > t_{n,2} - cn$ is simple (a blow-up of a bounded graph), and this bound is sharp.
• For $r \geq 3$, every $(r+1)$-saturated graph with $e(G) > t_{n,r} - (2 - \varepsilon)n/r$ is simple, and this bound is also sharp.
• The exact value of $\Lambda_r(r+2)$ is 2 for all $r \geq 3$, meaning that the extremal graphs for $\omega(G) = r$ and $\chi(G) \geq r+2$ are blow-ups of a graph on at most 2 vertices.
• The paper provides a new, short proof of Simonovits' theorem: every extremal $K_{r+1}$-free graph with $\chi(G) \geq k$ and $\omega(G) \leq r$ is a blow-up of a graph of order at most $m(k,r)$, with $m(k,r)$ absolute for fixed $k,r$.