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Packing subdivisions into regular graphs

Richard Montgomery, Kalina Petrova, Arjun Ranganathan, Jane Tan

TL;DR

This paper resolves Verstraëte's conjecture by showing that for any fixed graph $F$ and any $\eta>0$, there exists a threshold $d_0$ such that every $n$-vertex $d$-regular graph with $d\ge d_0$ contains a vertex-disjoint packing of $F$-subdivisions covering at least $(1-\eta)n$ vertices. The authors develop a probabilistic partitioning framework using the Lovász Local Lemma and Chernoff bounds, arrange the bulk of the graph into long vertex-disjoint paths, and use a small random subset to form an auxiliary graph in which Mader-type arguments guarantee the existence of $F$-subdivisions; a maximality argument then yields near-complete coverage. The approach extends Verstraëte's conjecture beyond the previously required polylogarithmic degree bound and clarifies the role of subdivided substructures in regular graphs. This advances the understanding of topological subgraph packings and has potential implications for balanced subdivisions and related packing problems in sparse graphs.

Abstract

We show that, for any graph $F$ and $η>0$, there exists a $d_0=d_0(F,η)$ such that every $n$-vertex $d$-regular graph with $d \geq d_0$ has a collection of vertex-disjoint $F$-subdivisions covering at least $(1-η)n$ vertices. This verifies a conjecture of Verstraëte from 2002 and improves a recent result of Letzter, Methuku and Sudakov which additionally required $d$ to be at least polylogarithmic in $n$.

Packing subdivisions into regular graphs

TL;DR

This paper resolves Verstraëte's conjecture by showing that for any fixed graph and any , there exists a threshold such that every -vertex -regular graph with contains a vertex-disjoint packing of -subdivisions covering at least vertices. The authors develop a probabilistic partitioning framework using the Lovász Local Lemma and Chernoff bounds, arrange the bulk of the graph into long vertex-disjoint paths, and use a small random subset to form an auxiliary graph in which Mader-type arguments guarantee the existence of -subdivisions; a maximality argument then yields near-complete coverage. The approach extends Verstraëte's conjecture beyond the previously required polylogarithmic degree bound and clarifies the role of subdivided substructures in regular graphs. This advances the understanding of topological subgraph packings and has potential implications for balanced subdivisions and related packing problems in sparse graphs.

Abstract

We show that, for any graph and , there exists a such that every -vertex -regular graph with has a collection of vertex-disjoint -subdivisions covering at least vertices. This verifies a conjecture of Verstraëte from 2002 and improves a recent result of Letzter, Methuku and Sudakov which additionally required to be at least polylogarithmic in .

Paper Structure

This paper contains 8 sections, 7 theorems, 10 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Proof outline
  3. Further discussion
  4. Preliminaries
  5. Partitioning and the local lemma
  6. Partitioning into paths
  7. Proof of Theorem \ref{['thm:packing']}
  8. Acknowledgements.

Key Result

Theorem 1.1

For any $F$ and $\eta>0$, there is some $d_0$ such that the following holds for each $n\geq d\geq d_0$. Every $d$-regular $n$-vertex graph contains a vertex-disjoint collection of subdivisions of $F$ covering at least $(1-\eta)n$ vertices.

Figures (1)

  • Figure 1: Our proof strategy with $F=K_4-e$ and $V(G)$ partitioned into $V$ and $W$, where most of $G[V]$ is partitioned into the collection $\mathcal{P}$ of paths. Subdivisions of $F$ in previous iterations of the auxiliary graph $L$ are drawn in bold, and occupy $W\setminus U$. Each bold edge corresponds to a distinct path in $V$ shown in red (we only show some of these correspondences, indicated by a dashed line). The current auxiliary graph $L$ is drawn, with vertex set $U'\subset U$.

Theorems & Definitions (12)

  • Theorem 1.1
  • Theorem 1.2: Mader mader1967homomorphieeigenschaften
  • Lemma 2.1: Lovász local lemma
  • Lemma 2.2: Chernoff's bound
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • proof
  • Lemma 3.1
  • proof : Proof of \ref{['thm:packing']}
  • ...and 2 more