On the typical structure of graphs not containing a fixed vertex-critical subgraph

TL;DR

This paper proves that the typical structure of sparse graphs avoiding a fixed vertex-critical subgraph undergoes a sharp phase transition, extending previous results from specific cases like odd cycles to a broader class of graphs.

Contribution

It resolves a conjecture by showing that a phase transition in the structure of H-free graphs occurs for all vertex-critical graphs, generalizing prior work on specific graph families.

Findings

Established a sharp threshold for the structural phase transition in H-free graphs.

Extended the phase transition phenomenon from odd cycles to all vertex-critical graphs.

Confirmed the conjecture for a broad class of graphs including all edge-critical graphs.

Abstract

This work studies the typical structure of sparse $H$-free graphs, that is, graphs that do not contain a subgraph isomorphic to a given graph $H$. Extending the seminal result of Osthus, Pr\"omel, and Taraz that addressed the case where $H$ is an odd cycle, Balogh, Morris, Samotij, and Warnke proved that, for every $r \ge 3$, the structure of a random $K_{r+1}$-free graph with $n$ vertices and $m$ edges undergoes a phase transition when $m$ crosses an explicit (sharp) threshold function $m_r(n)$. They conjectured that a similar threshold phenomenon occurs when $K_{r+1}$ is replaced by any strictly $2$-balanced, edge-critical graph $H$. In this paper, we resolve this conjecture. In fact, we prove that the structure of a typical $H$-free graph undergoes an analogous phase transition for every $H$ in a family of vertex-critical graphs that includes all edge-critical graphs.