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Nearly Hamilton cycles in sublinear expanders, and applications

Shoham Letzter, Abhishek Methuku, Benny Sudakov

TL;DR

The authors develop a novel toolkit for constructing nearly Hamilton cycles in sublinear expanders and for finding such expanders inside general graphs. Central to the approach is a refining procedure that yields expanders with strong regularity and a probabilistic framework that connects vertex pairs through random subsets, enabling almost-spanning F-subdivisions. They apply these methods to confirm a strong, polylogarithmic-degree version of Verstraëte’s subdivision-packing conjecture and derive several corollaries on cycle partitions and chord-rich cycles in regular graphs. The work links deterministic sparse expanders with embedding problems in dense hosts, offering new tools for sparse graph embeddings and potential broad applicability to related packing and subdivision questions.

Abstract

We develop novel methods for constructing nearly Hamilton cycles in sublinear expanders with good regularity properties, as well as new techniques for finding such expanders in general graphs. These methods are of independent interest due to their potential for various applications to embedding problems in sparse graphs. In particular, using these tools, we make substantial progress towards a twenty-year-old conjecture of Verstraëte, which asserts that for any given graph $F$, nearly all vertices of every $d$-regular graph $G$ can be covered by vertex-disjoint $F$-subdivisions. This significantly extends previous work on the conjecture by Kelmans, Mubayi and Sudakov, Alon, and Kühn and Osthus. Additionally, we present applications of our methods to two other problems.

Nearly Hamilton cycles in sublinear expanders, and applications

TL;DR

The authors develop a novel toolkit for constructing nearly Hamilton cycles in sublinear expanders and for finding such expanders inside general graphs. Central to the approach is a refining procedure that yields expanders with strong regularity and a probabilistic framework that connects vertex pairs through random subsets, enabling almost-spanning F-subdivisions. They apply these methods to confirm a strong, polylogarithmic-degree version of Verstraëte’s subdivision-packing conjecture and derive several corollaries on cycle partitions and chord-rich cycles in regular graphs. The work links deterministic sparse expanders with embedding problems in dense hosts, offering new tools for sparse graph embeddings and potential broad applicability to related packing and subdivision questions.

Abstract

We develop novel methods for constructing nearly Hamilton cycles in sublinear expanders with good regularity properties, as well as new techniques for finding such expanders in general graphs. These methods are of independent interest due to their potential for various applications to embedding problems in sparse graphs. In particular, using these tools, we make substantial progress towards a twenty-year-old conjecture of Verstraëte, which asserts that for any given graph , nearly all vertices of every -regular graph can be covered by vertex-disjoint -subdivisions. This significantly extends previous work on the conjecture by Kelmans, Mubayi and Sudakov, Alon, and Kühn and Osthus. Additionally, we present applications of our methods to two other problems.

Paper Structure

This paper contains 29 sections, 24 theorems, 7 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Long cycles in Expanders
  3. Packing subgraphs in regular graphs
  4. Organization of the paper
  5. Proof sketch
  6. A refining procedure.
  7. Preliminaries
  8. Notation
  9. Expansion
  10. Probabilistic tools
  11. A regularisation theorem
  12. Packing a regular graph with nearly regular expanders
  13. Packing with very weak expanders
  14. Refining procedure
  15. Connecting vertex pairs through a random vertex subset of an expander
  16. ...and 14 more sections

Key Result

Theorem 1.2

For any graph $F$ and large enough $n$, every $d$-regular graph $G$ of order $n$ with $d \ge (\log n)^{130}$ contains a $\mathop{\mathrm{TF}}\nolimits$-packing that covers all but at most $\frac{n}{(\log \log n)^{1/30}}$ vertices of $G$.

Figures (2)

  • Figure 1: If we cannot find sufficiently many connecting paths of length two (shown in blue), we identify a set $S$ of leaves that expands very well into $V_0$ and connect vertices in $S$ using paths (shown in red) through $V_0$.
  • Figure 2: The figure shows how to construct an almost spanning $F$-subdivision in $H$ (when $F = K_4$).

Theorems & Definitions (32)

  • Conjecture 1.1: Verstraëte verstraete2002note, 2002
  • Theorem 1.2
  • Lemma 2.1
  • Lemma 2.2
  • Definition 3.1: $\lambda$-expander
  • Definition 3.2: $(\varepsilon, c, s)$-expander
  • Lemma 3.3
  • Theorem 3.4: Chernoff's bound
  • Lemma 3.5
  • Theorem 3.6
  • ...and 22 more