An asymmetric container lemma and the structure of graphs with no induced $4$-cycle

TL;DR

This paper refines the hypergraph container method to analyze the structure of graphs with no induced 4-cycle, revealing they are typically partitioned into almost-independent and almost-clique parts, with phase transitions in random models.

Contribution

It introduces an asymmetric container lemma tailored for hereditary properties, enabling the characterization of typical graphs avoiding induced 4-cycles and identifying phase transitions in conditioned random graphs.

Findings

Almost all such graphs have a partition into an almost-independent set and an almost-clique.

The structure result applies for m satisfying n^{4/3} (log n)^4 ≤ m ≪ n^2.

Phase transitions in G(n,p) conditioned on no induced 4-cycle occur at p = n^{-2/3+o(1)} and p = n^{-1/3+o(1)}.

Abstract

The method of hypergraph containers, introduced recently by Balogh, Morris, and Samotij, and independently by Saxton and Thomason, has proved to be an extremely useful tool in the study of various monotone graph properties. In particular, a fairly straightforward application of this technique allows one to locate, for each non-bipartite graph $H$, the threshold at which the distribution of edges in a typical $H$-free graph with a given number of edges undergoes a transition from 'random-like' to 'structured'. On the other hand, for non-monotone hereditary graph properties the standard version of this method does not allow one to establish even the existence of such a threshold. In this paper we introduce a refinement of the container method that takes into account the asymmetry between edges and non-edges in a sparse member of a hereditary graph property. As an application, we determine the approximate structure of a typical graph with $n$ vertices, $m$ edges, and no induced copy of the $4$-cycle, for each function $m = m(n)$ satisfying $n^{4/3} (\log n)^4 \leqslant m \ll n^2$. We show that almost all such graphs $G$ have the following property: the vertex set of $G$ can be partitioned into an 'almost-independent' set (a set with $o(m)$ edges) and an 'almost-clique' (a set inducing a subgraph with density $1-o(1)$). The lower bound on $m$ is optimal up to a polylogarithmic factor, as standard arguments show that if $n \ll m \ll n^{4/3}$, then almost all such graphs are 'random-like'. As a further consequence, we deduce that the random graph $G(n,p)$ conditioned to contain no induced $4$-cycles undergoes phase transitions at $p = n^{-2/3 + o(1)}$ and $p = n^{-1/3 + o(1)}$.