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

Abstract

AbstractThis work studies the typical structure of sparse ‐free graphs, that is, graphs that do not contain a subgraph isomorphic to a given graph . Extending the seminal result of Osthus, Prömel, and Taraz that addressed the case where is an odd cycle, Balogh, Morris, Samotij, and Warnke proved that, for every , the structure of a random ‐free graph with vertices and edges undergoes a phase transition when crosses an explicit (sharp) threshold function . They conjectured that a similar threshold phenomenon occurs when is replaced by any strictly 2‐balanced, edge‐critical graph . In this paper, we resolve this conjecture. In fact, we prove that the structure of a typical ‐free graph undergoes an analogous phase transition for every in a family of vertex‐critical graphs that includes all edge‐critical graphs.