[Paper Review] The typical structure of graphs with no large cliques

József Balogh, Neal Bushaw+4 more|arXiv (Cornell University)|Jun 26, 2014

Limits and Structures in Graph Theory16 references4 citations

TL;DR

This paper establishes that for all functions $ r = r(n) $ with $ r \leq (\log n)^{1/4} $, almost all $ K_{r+1} $-free graphs on $ n $ vertices are $ r $-partite. The proof combines a new supersaturation result for the Erdős–Simonovits stability theorem, the hypergraph container method, and a counting technique to show that graphs far from being $ r $-partite are exponentially rare, thus proving the typical structure is $ r $-partite even as $ r $ grows with $ n $.

ABSTRACT

In 1987, Kolaitis, Prömel and Rothschild proved that, for every fixed $r \in \mathbb{N}$, almost every $n$-vertex $K_{r+1}$-free graph is $r$-partite. In this paper we extend this result to all functions $r = r(n)$ with $r \leqslant (\log n)^{1/4}$. The proof combines a new (close to sharp) supersaturation version of the Erdős-Simonovits stability theorem, the hypergraph container method, and a counting technique developed by Balogh, Bollobás and Simonovits.

Motivation & Objective

• Extend the classical result that almost all $ K_{r+1} $-free graphs are $ r $-partite from fixed $ r $ to functions $ r = r(n) $ growing with $ n $.
• Address the challenge of characterizing the typical structure of $ K_{r+1} $-free graphs when the forbidden clique size increases with $ n $, particularly in the regime where $ r \leq (\log n)^{1/4} $.
• Establish a sharp supersaturation result for the Erdős–Simonovits stability theorem to control the number of $ K_{r+1} $-free graphs that are far from being $ r $-partite.
• Combine the hypergraph container method with a refined counting technique to show that such non-$ r $-partite graphs are exponentially rare compared to $ r $-partite ones.
• Provide a quantitative structure theorem for $ K_{r+1} $-free graphs that strengthens the understanding of extremal and typical graph properties in the sparse, growing-clique setting.

Proposed method

• Introduce a new supersaturation result (Theorem 1.2) that gives a nearly optimal lower bound on the number of $ K_{r+1} $ copies in graphs $ t $-far from being $ r $-partite, with the bound being sharp up to a factor of $ e^r $.
• Apply the hypergraph container method to control the number of $ K_{r+1} $-free graphs that are not $ r $-partite, by encoding them via containers that avoid large cliques.
• Use a refined counting technique from Balogh, Bollobás, and Simonovits to compare the number of $ K_{r+1} $-free graphs that are close to $ r $-partite with those that are not.
• Define and analyze the set $ \mathcal{Q}(n,r) $, consisting of $ K_{r+1} $-free graphs that are $ n^{2-1/r^2} $-close to being $ r $-partite but not $ r $-partite, to isolate the exceptional cases.
• Bound the size of the set of optimal partitions for graphs in $ \mathcal{Q}(n,r) $ using entropy and binomial coefficient estimates, showing it is subexponential in $ n $.
• Combine all bounds via double counting in the container graph $ F_m $, showing that $ |\mathcal{Q}(n,r)| $ is exponentially smaller than $ |\mathcal{K}(n,r)| $, the number of $ r $-partite $ K_{r+1} $-free graphs.

Experimental results

Research questions

• RQ1 Can the classical result that almost all $ K_{r+1} $-free graphs are $ r $-partite be extended to the case where $ r $ grows with $ n $, rather than being fixed?
• RQ2 What is the maximum growth rate of $ r(n) $ for which almost all $ K_{r+1} $-free graphs remain $ r $-partite?
• RQ3 How can one quantify the number of $ K_{r+1} $-free graphs that are far from being $ r $-partite, especially when $ r $ grows with $ n $?
• RQ4 Is there a supersaturation result for the Erdős–Simonovits stability theorem that is strong enough to control the number of non-$ r $-partite $ K_{r+1} $-free graphs in the growing $ r $ regime?
• RQ5 Can the hypergraph container method be combined with refined counting to show that the exceptional graphs (non-$ r $-partite but $ K_{r+1} $-free) are negligible in number compared to $ r $-partite ones?

Key findings

• Theorem 1.1 establishes that for all $ r = r(n) \leq (\log n)^{1/4} $, almost all $ K_{r+1} $-free graphs on $ n $ vertices are $ r $-partite, extending the classical result to growing $ r $.
• A new supersaturation result (Theorem 1.2) shows that any graph $ t $-far from being $ r $-partite contains at least $ \frac{n^{r-1}}{e^{2r} r!} \left( e(G) + t - \left(1 - \frac{1}{r}\right)\frac{n^2}{2} \right) $ copies of $ K_{r+1} $, which is sharp up to a factor of $ e^r $.
• The number of $ K_{r+1} $-free graphs that are $ n^{2-1/r^2} $-close to being $ r $-partite but not $ r $-partite is at most $ 2^{-2^{-6r}n} \cdot |\mathcal{K}(n,r)| $, showing they are exponentially rare.
• By combining the container method with the new supersaturation result and counting techniques, the authors prove that the set $ \mathcal{Q}(n,r) $ of such exceptional graphs is negligible compared to the set of $ r $-partite $ K_{r+1} $-free graphs.
• The bound $ r \leq (\log n)^{1/4} $ is nearly optimal, as the threshold for the typical structure to shift is expected to be around $ r \sim \log n $, and the result is within a constant factor of this.