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Packing subgraphs in regular graphs

Shoham Letzter, Abhishek Methuku, Benny Sudakov

TL;DR

This work settles two long-standing problems on packing in dense regular graphs. It proves that for every bipartite graph $H$ and every $c>0$, every $ig floor cn ig floor$-regular graph $G$ on $n$ vertices contains an $H$-packing that misses only a constant number of vertices, answering a 2005 Kühn–Osthus question; it further shows that the vertices can be almost perfectly packed with subdivisions of $K_t$ for any fixed $tigge 2$, resolving Verstraëte’s 2002 conjecture in this regime. The core method blends balancing expanders and super-regular subgraphs with robust expanders, the regularity lemma, and the blow-up lemma. The approach reduces complex packings to near-perfect $K_{t,t}$-packings in balanced or far-from-bipartite expanders, then lifts these packings through a fractional-to-integral conversion via Hamiltonicity and absorption techniques. Overall, the results advance the understanding of how high regularity forces near-complete tilings and topological packings in dense graphs, with potential extensions to hypergraphs and monochromatic variants.

Abstract

An $H$-packing in a graph $G$ is a collection of pairwise vertex-disjoint copies of $H$ in $G$. We prove that for every $c > 0$ and every bipartite graph $H$, any $\lfloor{cn}\rfloor$-regular graph $G$ admits an $H$-packing that covers all but a constant number of vertices. This resolves a problem posed by Kühn and Osthus in 2005. Moreover, our result is essentially tight: the conclusion fails if $G$ is not both regular and sufficiently dense; it is in general not possible to guarantee covering all vertices of $G$ by an $H$-packing, and if $H$ is not bipartite then $G$ need not contain any copies of $H$. We also prove that for all $c > 0$, integers $t \geq 2$, and sufficiently large $n$, all the vertices of every $\lfloor cn \rfloor$-regular graph can be covered by vertex-disjoint subdivisions of $K_t$. This resolves another problem of Kühn and Osthus from 2005, which goes back to a conjecture of Verstraëte from 2002. Our proofs combine novel methods for balancing expanders and super-regular subgraphs with a number of powerful techniques including properties of robust expanders, the regularity lemma, and the blow-up lemma.

Packing subgraphs in regular graphs

TL;DR

This work settles two long-standing problems on packing in dense regular graphs. It proves that for every bipartite graph and every , every -regular graph on vertices contains an -packing that misses only a constant number of vertices, answering a 2005 Kühn–Osthus question; it further shows that the vertices can be almost perfectly packed with subdivisions of for any fixed , resolving Verstraëte’s 2002 conjecture in this regime. The core method blends balancing expanders and super-regular subgraphs with robust expanders, the regularity lemma, and the blow-up lemma. The approach reduces complex packings to near-perfect -packings in balanced or far-from-bipartite expanders, then lifts these packings through a fractional-to-integral conversion via Hamiltonicity and absorption techniques. Overall, the results advance the understanding of how high regularity forces near-complete tilings and topological packings in dense graphs, with potential extensions to hypergraphs and monochromatic variants.

Abstract

An -packing in a graph is a collection of pairwise vertex-disjoint copies of in . We prove that for every and every bipartite graph , any -regular graph admits an -packing that covers all but a constant number of vertices. This resolves a problem posed by Kühn and Osthus in 2005. Moreover, our result is essentially tight: the conclusion fails if is not both regular and sufficiently dense; it is in general not possible to guarantee covering all vertices of by an -packing, and if is not bipartite then need not contain any copies of . We also prove that for all , integers , and sufficiently large , all the vertices of every -regular graph can be covered by vertex-disjoint subdivisions of . This resolves another problem of Kühn and Osthus from 2005, which goes back to a conjecture of Verstraëte from 2002. Our proofs combine novel methods for balancing expanders and super-regular subgraphs with a number of powerful techniques including properties of robust expanders, the regularity lemma, and the blow-up lemma.

Paper Structure

This paper contains 23 sections, 20 theorems, 3 equations.

Table of Contents

  1. Introduction
  2. $H$-packings in dense regular graphs
  3. Packing subdivisions in dense regular graphs
  4. Organisation of the paper
  5. Notation
  6. Sketch of the proof of \ref{['mainthm:packingsubgraphs']}
  7. Balancing the expanders
  8. Packing $K_{t,t}$'s in expanders
  9. Useful lemmas and tools
  10. Concentration inequality
  11. Expanders in dense regular graphs
  12. The regularity lemma and the blow-up lemma
  13. The Kővári--Sós--Turán theorem
  14. Balancing the expanders
  15. Packing $K_{t,t}$'s in expanders
  16. ...and 8 more sections

Key Result

Theorem 1.1

For every bipartite graph $H$ and every $0 < c, \alpha \le 1$, there exists $n_0 = n_0(H, c, \alpha)$ such that every $d$-regular graph $G$ of order $n$, where $d \ge cn$ and $n \ge n_0$, has an $H$-packing that covers all but at most $\alpha n$ vertices of $G$.

Theorems & Definitions (21)

  • Theorem 1.1: Kühn and Osthus kuhn2005packings
  • Theorem 1.3: Kühn and Osthus kuhn2005packings
  • Theorem 1.4
  • Conjecture 1.5: Verstraëte verstraete2002note
  • Theorem 1.7
  • Lemma 3.1: Theorem 2.10 in janson2000wiley
  • Lemma 3.2: Lemma 2.1 from gruslys2021cycle
  • Lemma 3.3
  • Lemma 3.4: Lemma 2.3 from gruslys2021cycle
  • Lemma 3.5: regularity lemma
  • ...and 11 more