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

Oren Engelberg, Wojciech Samotij, Lutz Warnke

TL;DR

The paper proves a sharp threshold phenomenon for the typical structure of H-free graphs in the sparse regime, extending known results from edge-critical to a broader class of vertex-critical graphs. It introduces a threshold m_H(n) defined via the 2-density m_2(H) and, for vertex-critical H with χ(H)=r+1 and criticality k+1, shows that when m ≥ C_H m_H, almost all graphs in F_{n,m}(H) become (χ(H)−1)-partite in the sense of belonging to G(r,k). The argument combines a detailed sparse-structure analysis of almost r-colorable graphs, probabilistic inequalities (Hypergeometric Janson and Harris), and a careful dense-case treatment to establish both the 0- and 1-statements, including an approximate structural description and an exact threshold for a broad family of H. This advances understanding of the transition from coarse, extremal structures to precise partitioning in sparse random H-free graphs and highlights the intricate role of 2-density, critical stars, and color-class distributions. The results have implications for extremal combinatorics and random graph theory by clarifying when sparse H-free graphs exhibit near-partite organization and how this depends on the underlying criticality of H.

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, Prömel, 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.

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

TL;DR

The paper proves a sharp threshold phenomenon for the typical structure of H-free graphs in the sparse regime, extending known results from edge-critical to a broader class of vertex-critical graphs. It introduces a threshold m_H(n) defined via the 2-density m_2(H) and, for vertex-critical H with χ(H)=r+1 and criticality k+1, shows that when m ≥ C_H m_H, almost all graphs in F_{n,m}(H) become (χ(H)−1)-partite in the sense of belonging to G(r,k). The argument combines a detailed sparse-structure analysis of almost r-colorable graphs, probabilistic inequalities (Hypergeometric Janson and Harris), and a careful dense-case treatment to establish both the 0- and 1-statements, including an approximate structural description and an exact threshold for a broad family of H. This advances understanding of the transition from coarse, extremal structures to precise partitioning in sparse random H-free graphs and highlights the intricate role of 2-density, critical stars, and color-class distributions. The results have implications for extremal combinatorics and random graph theory by clarifying when sparse H-free graphs exhibit near-partite organization and how this depends on the underlying criticality of H.

Abstract

This 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 -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.

Paper Structure

This paper contains 40 sections, 36 theorems, 311 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Background and motivation
  3. Our result -- edge-critical graphs
  4. Our result -- vertex-critical graphs
  5. Acknowledgements
  6. Outline of the proof
  7. Why is $m_H$ the threshold?
  8. The 0-statement
  9. The 1-statement
  10. Preliminaries
  11. Probabilistic inequalities
  12. Two-density related bounds
  13. The Turán problem for $r$-partite graphs
  14. Estimates for binomial coefficients
  15. On almost $r$-colourable graphs
  16. ...and 25 more sections

Key Result

Theorem 1.2

For every strictly $2$-balanced, non-bipartite, edge-critical graph $H$, there exist positive constants $c_H$ and $C_H$ such that, letting the following holds for a uniformly chosen random graph $F_{n,m} \in \mathcal{F}_{n,m}(H)$:

Figures (1)

  • Figure 1: The Hasse diagram depicting dependence between the various constants in the proof

Theorems & Definitions (79)

  • Conjecture 1.1: BaMoSaWa16
  • Theorem 1.2
  • Theorem 1.3: HuPrSt93
  • Remark
  • Definition
  • Remark
  • Remark
  • Theorem 1.4
  • Lemma 3.1: Hypergeometric Janson Inequality
  • Lemma 3.2: Hypergeometric Harris Inequality
  • ...and 69 more