Packing subgraphs in regular graphs
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arXiv
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| Format: | Preprint |
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2025
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| author | Letzter, Shoham Methuku, Abhishek Sudakov, Benny |
| author_facet | Letzter, Shoham Methuku, Abhishek Sudakov, Benny |
| contents | An \emph{$H$-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 non-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, regularity lemma, and blow-up lemma. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_26180 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Packing subgraphs in regular graphs Letzter, Shoham Methuku, Abhishek Sudakov, Benny Combinatorics An \emph{$H$-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 non-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, regularity lemma, and blow-up lemma. |
| title | Packing subgraphs in regular graphs |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2509.26180 |