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Extremal density for subdivisions with length or sparsity constraints

Jaehoon Kim, Hong Liu, Yantao Tang, Guanghui Wang, Donglei Yang, Fan Yang

TL;DR

This work resolves the question of whether a linear-in-$e(H)$ average-degree bound guarantees a balanced $H$-subdivision in any graph with no isolated vertices, establishing that $G$ contains $TH^{(\ell)}$ for some $\ell$ whenever $d(G)\ge C\,e(H)$. It further identifies a threshold phenomenon: for graphs $H$ with logarithmic density, a sublinear bound can suffice under a separability condition (biseparability), and the authors provide a sublogarithmic-density theorem in this regime. The core methodology weaves together sublinear expanders, dependent random choice, the regularity lemma, and novel anchor structures (units and webs) with adjusters to precisely control path lengths. These techniques enable embedding balanced subdivisions in both dense and sparse host graphs and yield implications for stochastic block models and Cartesian product graphs, advancing the understanding of subdivisions under density constraints with potential applications in topological and structural graph theory.

Abstract

Given a graph $H$, a balanced subdivision of $H$ is obtained by replacing all edges of $H$ with internally disjoint paths of the same length. In this paper, we prove that for any graph $H$, a linear-in-$e(H)$ bound on average degree guarantees a balanced $H$-subdivision. This strengthens an old result of Bollobás and Thomason, and resolves a question of Gil-Fernández, Hyde, Liu, Pikhurko and Wu. We observe that this linear bound on average degree is best possible whenever $H$ is logarithmically dense. We further show that this logarithmic density is the critical threshold: for many graphs $H$ below this density, its subdivisions are forcible by a sublinear-in-$e(H)$ bound on average degree. We provide such examples by proving that the subdivisions of any almost bipartite graph $H$ with sublogarithmic density are forcible by a sublinear-in-$e(H)$ bound on average degree, provided that $H$ satisfies some additional separability condition.

Extremal density for subdivisions with length or sparsity constraints

TL;DR

This work resolves the question of whether a linear-in- average-degree bound guarantees a balanced -subdivision in any graph with no isolated vertices, establishing that contains for some whenever . It further identifies a threshold phenomenon: for graphs with logarithmic density, a sublinear bound can suffice under a separability condition (biseparability), and the authors provide a sublogarithmic-density theorem in this regime. The core methodology weaves together sublinear expanders, dependent random choice, the regularity lemma, and novel anchor structures (units and webs) with adjusters to precisely control path lengths. These techniques enable embedding balanced subdivisions in both dense and sparse host graphs and yield implications for stochastic block models and Cartesian product graphs, advancing the understanding of subdivisions under density constraints with potential applications in topological and structural graph theory.

Abstract

Given a graph , a balanced subdivision of is obtained by replacing all edges of with internally disjoint paths of the same length. In this paper, we prove that for any graph , a linear-in- bound on average degree guarantees a balanced -subdivision. This strengthens an old result of Bollobás and Thomason, and resolves a question of Gil-Fernández, Hyde, Liu, Pikhurko and Wu. We observe that this linear bound on average degree is best possible whenever is logarithmically dense. We further show that this logarithmic density is the critical threshold: for many graphs below this density, its subdivisions are forcible by a sublinear-in- bound on average degree. We provide such examples by proving that the subdivisions of any almost bipartite graph with sublogarithmic density are forcible by a sublinear-in- bound on average degree, provided that satisfies some additional separability condition.
Paper Structure (25 sections, 32 theorems, 76 equations, 2 figures)

This paper contains 25 sections, 32 theorems, 76 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Balanced subdivisions
  3. When does a sublinear bound suffice?
  4. Related work
  5. Preliminaries
  6. Notation
  7. Tools
  8. Sublinear expander
  9. Proof of Proposition \ref{['random']}
  10. Main lemmas and overviews
  11. Proof of Theorem \ref{['main1']}
  12. Proof overview of Lemma \ref{['dense']}
  13. Proofs of Theorems \ref{['thm: sublog']} and \ref{['ans2-pretty-version']}
  14. Proof of Lemma \ref{['dense']}
  15. Reduction to locally dense graphs
  16. ...and 10 more sections

Key Result

Theorem 1.1

There exists a constant $C>0$ such that, for any graph $H$ with no isolated vertices, if $G$ is a graph with average degree $d(G)\geq C\cdot e(H)$, then $G$ contains a $TH^{(\ell)}$ for some $\ell\in \mathbb{N}$.

Figures (2)

  • Figure 1: Embedding
  • Figure 2: $(h_0,h_1,h_2,h_3)$-web

Theorems & Definitions (87)

  • Theorem 1.1
  • Proposition 1.2
  • Definition 1.3: Biseparable
  • Theorem 1.4
  • Corollary 1.5
  • Theorem 1.6
  • Lemma 2.1: KST, Kővári-Sós-Turán
  • Lemma 2.2: Lee
  • Definition 2.3: $\varepsilon$-regular pair
  • Lemma 2.4: RD
  • ...and 77 more