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Supersaturation via edge-gluing

Zihao Jin, Sean Longbrake, Liana Yepremyan

TL;DR

The paper develops a robust framework for edge- and subgraph-gluing in bipartite graphs to preserve Erdős–Simonovits supersaturation properties. Central to the approach is a cleaning lemma that distributes embeddings evenly and provides precise control over how many copies of a fixed subgraph participate in a given edge, vertex, or substructure, enabling reductions to $K$-almost-regular hosts and probabilistic counting. The authors show that gluing two supersaturated graphs along a fixed edge yields a new graph that remains supersaturated (under symmetry conditions), and that gluing multiple copies along a subforest preserves a weaker but still meaningful supersaturation bound; they also obtain sharp results for cycle-blowups of trees and related constructions. Collectively, these results extend Sidorenko-type closure properties to ES-conjectures and furnish a versatile toolkit for constructing new supersaturated graphs in sparse regimes, with implications for Turán-type extremal problems and sparse graph counting. The work provides explicit exponent updates $\alpha'$ and constants governing the extremal and saturation behavior of glued graphs, linking classical extremal theory to modern supersaturation phenomena.

Abstract

In 1984, Erdős and Simonovits conjectured the following: given a bipartite graph $H$, there exist constants $β, C > 0$ such that any graph $G$ on $n$ vertices and $pn^2\geq C \mathrm{ex}(n, H)$ edges contains at least $βn^{\mathrm{v}(H)} p^{\mathrm{e}(H)}$ copies of $H$. We show that edge-gluing preserves the satisfiability of this conjecture under some mild symmetry conditions. Namely, if two graphs $H_1$ and $H_2$ satisfy this conjecture, and if furthermore, gluing them along a fixed edge produces a unique graph then the resulting graph satisfies the conjecture as well. In the same paper, Erdős and Simonovits conjectured a weaker statement: for every $H$, there is some $α, β, C > 0$ such that any graph $G$ on $n$ vertices and $pn^2\geq C n^{1+ α}$ edges contains at least $βn^{\mathrm{v}(H)} p^{\mathrm{e}(H)}$ copies of $H$. We show that if $H$ satisfies this conjecture then by gluing several copies of labeled $H$ along the same copy of a subforest of $H$ produces a graph that also satisfies the conjecture.

Supersaturation via edge-gluing

TL;DR

The paper develops a robust framework for edge- and subgraph-gluing in bipartite graphs to preserve Erdős–Simonovits supersaturation properties. Central to the approach is a cleaning lemma that distributes embeddings evenly and provides precise control over how many copies of a fixed subgraph participate in a given edge, vertex, or substructure, enabling reductions to -almost-regular hosts and probabilistic counting. The authors show that gluing two supersaturated graphs along a fixed edge yields a new graph that remains supersaturated (under symmetry conditions), and that gluing multiple copies along a subforest preserves a weaker but still meaningful supersaturation bound; they also obtain sharp results for cycle-blowups of trees and related constructions. Collectively, these results extend Sidorenko-type closure properties to ES-conjectures and furnish a versatile toolkit for constructing new supersaturated graphs in sparse regimes, with implications for Turán-type extremal problems and sparse graph counting. The work provides explicit exponent updates and constants governing the extremal and saturation behavior of glued graphs, linking classical extremal theory to modern supersaturation phenomena.

Abstract

In 1984, Erdős and Simonovits conjectured the following: given a bipartite graph , there exist constants such that any graph on vertices and edges contains at least copies of . We show that edge-gluing preserves the satisfiability of this conjecture under some mild symmetry conditions. Namely, if two graphs and satisfy this conjecture, and if furthermore, gluing them along a fixed edge produces a unique graph then the resulting graph satisfies the conjecture as well. In the same paper, Erdős and Simonovits conjectured a weaker statement: for every , there is some such that any graph on vertices and edges contains at least copies of . We show that if satisfies this conjecture then by gluing several copies of labeled along the same copy of a subforest of produces a graph that also satisfies the conjecture.

Paper Structure

This paper contains 14 sections, 17 theorems, 41 equations.

Table of Contents

  1. Introduction
  2. Notation and Preliminary results
  3. Proof:
  4. Proof:
  5. Proof:
  6. Proof:
  7. Proofs of Main Results
  8. Proof:
  9. Proof:
  10. Proof:
  11. Proof:
  12. Proof:
  13. Proof:
  14. Acknowledgments

Key Result

Theorem 1.1

Let $H_1$ and $H_2$ be two bipartite graphs with $\mathrm{ex}(n, H_1)~=~O(n^{1 + \alpha})$ and $\mathrm{ex}(n, H_2)~=~O(n^{1 + \beta})$. Fix $e_1 \in E(H_1)$ and $e_2 \in E(H_2)$. Then, the family $\mathcal{H}$ of graphs formed by identifying $e_1$ and $e_2$ satisfies

Theorems & Definitions (24)

  • Theorem 1.1: Lemma 4.1 in kupavskii2022extremal
  • Theorem 1.2: Theorem 1.5 in dong2025bipartite
  • Conjecture 1.3: sidorenko1991inequalities
  • Theorem 1.4: Theorem 2 in li2011logarithimic
  • Conjecture 1.5: Conjecture $2$ in ES-cube
  • Conjecture 1.6: Conjecture $2^*$ in ES-cube
  • Theorem 1.7
  • Theorem 1.8
  • Theorem 1.9
  • Corollary 1.10
  • ...and 14 more