Nearly Hamilton cycles in sublinear expanders, and applications

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.

💡 Research Summary

The paper “Nearly Hamilton cycles in sublinear expanders, and applications” develops a new framework for constructing almost‑Hamiltonian structures inside graphs that exhibit only sublinear expansion, i.e., an expansion factor α = Θ(1/(log n)^c) for some constant c>0. Classical Hamiltonicity results such as Dirac’s theorem or the Krivelevich‑Sudakov theorem apply to dense or constant‑expander graphs, but they give no guarantee of long cycles in this much sparser regime. The authors overcome this limitation by coupling two novel ingredients: (1) a method for extracting from any d‑regular graph (with d ≥ 2 log n) a collection of vertex‑disjoint subgraphs that are both sublinear expanders and highly regular (maximum degree d and average degree d·(1−o(1))), and (2) a technique for embedding a nearly spanning Hamiltonian path inside each such expander and then turning that path into an almost‑spanning subdivision of an arbitrary fixed graph F.

The first ingredient is formalised in Lemma 4.1 (and its precise version Lemma 4.1). By random sampling and a boot‑strapping argument the authors show that, after deleting a tiny set of vertices, the remaining graph retains the required expansion properties while its degree distribution becomes tightly concentrated around d. This yields a family ℋ of sublinear expanders covering all but o(n) vertices of the original regular graph. Corollary 4.2 extends the result to any graph whose average degree is Ω(d log n), showing that the construction is robust beyond regular inputs.

The second ingredient is built in two steps. Lemma 5.1 proves that, given a collection of vertex‑disjoint pairs satisfying a mild expansion condition, one can connect each pair by a short vertex‑disjoint path that passes through a randomly chosen vertex set R inside a sublinear expander. The randomness guarantees that the paths remain disjoint and that R supplies enough “bridges”. Lemma 6.1 then shows that a sufficiently regular sublinear expander contains a nearly Hamiltonian path P and that, using P as a backbone, one can embed an almost‑spanning subdivision of any fixed graph F. The embedding respects vertex‑disjointness of the internal vertices of the subdivision, which is crucial for the final packing argument.

Combining these tools, Theorem 1.2 is proved: for any fixed graph F and any sufficiently large n, every d‑regular graph G with d ≥ (log n)^130 contains a collection of vertex‑disjoint F‑subdivisions that cover all but at most n·(log log n)^{1/30} vertices. This resolves Verstraëte’s 2002 conjecture up to a polylogarithmic error term and dramatically improves earlier partial results that required either dense graphs or much larger uncovered sets.

Beyond the main theorem, the paper presents two further applications. First, it strengthens a conjecture of Magnant and Martin (2009) concerning the decomposition of d‑regular graphs into at most n/(d+1) vertex‑disjoint paths. Using the same machinery, the authors show that for sufficiently large degree the vertices can actually be partitioned into n/(d+1) cycles, i.e., a near‑perfect cycle decomposition. Second, they address a question of Chen, Erdős and Staton (1996) about cycles with many chords. By adapting the chord‑insertion technique within a sublinear expander, they recover (up to a slightly weaker polylogarithmic factor) a recent result of Draganić et al. on the existence of such highly chorded cycles.

Overall, the work demonstrates that sublinear expanders—despite their weak expansion—can be harnessed, when coupled with careful regularity control, to emulate many of the powerful embedding and packing results previously known only for dense or constant‑expander graphs. The methods introduced are likely to find further use in sparse graph embedding, decomposition, and extremal problems.