Packing subgraphs in regular graphs

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.

💡 Research Summary

The paper addresses two long‑standing open problems concerning packings in dense regular graphs. The first problem, posed by Kühn and Osthus in 2005, asks whether for every bipartite graph (H) and any constant (c>0) there exists a constant (C=C(H,c)) such that every (\lfloor cn\rfloor)-regular graph (G) on (n) vertices contains an (H)-packing covering all but at most (C) vertices. The authors answer this affirmatively (Theorem 1.4). Their proof proceeds by decomposing the regular graph into a bounded number of robust expanders using a structural decomposition of Gruslys and Letzter. Each expander may be close to bipartite or far from bipartite. A novel “balancing” procedure is introduced: by removing a carefully chosen small (K_{t,t})-packing, the authors transform imbalanced bipartite expanders into balanced ones, while ensuring that expanders that are far from bipartite remain so. After balancing, the problem reduces to finding an almost‑perfect (K_{t,t})-packing either in a balanced bipartite expander or in a non‑bipartite expander.

For the balanced case, the Szemerédi regularity lemma yields a collection of vertex‑disjoint super‑regular pairs covering almost all vertices. A fractional matching in the reduced graph, derived from the Hamiltonicity of the underlying expanders, is used to construct edge‑disjoint odd cycles that cover the reduced graph. The clusters are then split, a few vertices are removed to obtain perfectly super‑regular pairs, and a small (K_{t,t})-packing is used to absorb the exceptional vertices. A second small (K_{t,t})-packing is built to make the sizes of the two sides of each super‑regular pair divisible by (t). Finally, a carefully designed balancing (K_{t,t})-packing removes the remaining size discrepancy, allowing the blow‑up lemma to embed a perfect (K_{t,t})-packing in each pair.

The second problem, originally conjectured by Verstraëte (2002) and later refined by Kühn and Osthus, asks whether every dense regular graph can be packed with vertex‑disjoint subdivisions of an arbitrary graph (F). The authors prove a stronger statement (Theorem 1.7): for any graph (F) and any constant (c>0), every (\lfloor cn\rfloor)-regular graph on sufficiently many vertices admits a perfect (F)-subdivision packing. The proof follows a similar decomposition into robust expanders. Using a lemma from Gruslys–Letzter, a small linear forest (H) is found that balances the expanders, ensuring each expander contains either zero or two leaves of (H). The forest is then modified so that each expander receives at most one component, with both leaves inside the same expander. Inside each expander the authors construct two vertex‑disjoint (F)-subdivisions whose union remains balanced. Finally, the components of (H) and any leftover vertices are absorbed into these subdivisions using the Hamiltonicity and robust connectivity of the expanders, which guarantee short paths avoiding a prescribed small forbidden set.

The paper also discusses tightness: regularity and linear degree are necessary, as shown by constructions where non‑regular or sparse graphs fail to admit the desired packings, and where non‑bipartite (H) cannot appear in bipartite regular graphs. Moreover, the constant (C) in Theorem 1.4 must depend on both (c) and (H); otherwise counterexamples based on disjoint cliques of size (|V(H)|-1) force many uncovered vertices.

Methodologically, the work combines four powerful tools—robust expanders, the regularity lemma, the blow‑up lemma, and a new balancing technique—into a unified framework. The balancing of expanders, both in the bipartite and non‑bipartite settings, is a key novelty that differs from earlier approaches to Hamilton cycles in regular graphs. The fractional matching and odd‑cycle construction in the reduced graph provide a flexible way to handle divisibility constraints without requiring a perfect matching. The absorption arguments rely on the fact that any two vertices in a robust expander can be linked by a short path that avoids a small set, a property that is crucial for both the (K_{t,t}) and subdivision packings.

In summary, the authors resolve two major conjectures concerning almost‑perfect bipartite packings and perfect subdivision packings in dense regular graphs. Their results are essentially optimal, and the techniques introduced are likely to have further applications in extremal graph theory, particularly in problems where regularity and expansion properties can be exploited to achieve near‑perfect or perfect decompositions.