Hamiltonicity of regular sublinear expanders

Abstract: We say that a $d$-regular graph is a $γ$-expander if for every not too large set of vertices $S$, there are at least $γd |S|$ edges leaving $S$, and we say that a graph $G$ is $γ$-far from bipartite if at least $γe(G)$ edges need to be removed to make it bipartite. We prove that there exists an absolute constant $K$ such that any $n$-vertex $d$-regular $γ$-expander with $d \ge (γ^{-1} \log n)<sup>K$ is Hamiltonian, provided that it is bipartite or $γ$-far from bipartite. As applications, we obtain highly robust versions of recent important results on the Hamiltonicity of Cayley graphs and Kneser graphs. As part of our proof, we prove a random connecting lemma for sublinear expanders which might be of independent interest.

• The paper demonstrates that regular sublinear expanders with sufficiently large degrees ($d ime {(\gamma^{-1}\varepsilon^{-1}\log n)^{10^8}}$)
• The findings hold for bipartite expanders if $d > (\gamma^{-1}\log n)^{(10^8)}$ and for non-bipartite expanders if $d ≥ (\gamma^{-1}ε^{-1}\log n)^{10^8}$
• The Hamiltonian nature relies on the absorbing lemma that ensures that every single vertex and edge attaches properly to form a Hamiltonian Circuit that considers random gamble for the vertex

The paper proves that regular sublinear expanders of sufficiently large degree are Hamiltonian, provided they are either bipartite or robustly far from bipartite. This extends the recent Hamiltonicity theory of linear expanders — culminating in the resolution of the Krivelevich–Sudakov conjecture by Draganić, Montgomery, Munhá Correia, Pokrovskiy and Sudakov (Draganić et al., 2024) — to the much broader class of sublinear expanders introduced by Komlós and Szemerédi and refined in robust forms by Haslegrave–Kim–Liu and Sudakov–Tomon. The degree requirement is polylogarithmic: for a $d$-regular $\gamma$-expander that is $\varepsilon$-far from bipartite, Hamiltonicity holds whenever $d \ge (\gamma^{-1}\varepsilon^{-1}\log n)^{10^8}$.

Definitions and main results

An $n$-vertex graph $G$ is a $\gamma$-expander if every $S \subseteq V(G)$ with $1 \le |S| \le \tfrac{2}{3}n$ satisfies $e_G(S, V(G)\setminus S) \ge \gamma\bar{d}(G)|S|$, where $\gamma$0 may decay with $\gamma$1 (a prototypical value is $\gamma$2). The bipartite case is treated first: any bipartite $\gamma$3-regular $\gamma$4-expander with $\gamma$5 is Hamiltonian. In the non-bipartite case, a graph is $\gamma$6-far from bipartite if every cut contains at most $\gamma$7 edges; then $\gamma$8 suffices.

Both statements admit spectral formulations via Cheeger-type inequalities. For the far-from-bipartite version, since every $\gamma$9-regular graph is $\varepsilon$0-far from bipartite with $\varepsilon$1, one obtains: any $\varepsilon$2-graph with $\varepsilon$3 and $\varepsilon$4 is Hamiltonian. This is strictly stronger than prior work of Krivelevich–Sudakov, Glock–Munhá Correia–Sudakov, and Ferber–Han–Mao–Vershynin in that it allows $\varepsilon$5 arbitrarily close to 1, though it requires $\varepsilon$6 to be polylogarithmic; conversely, when $\varepsilon$7 is small, the constant-ratio theorem of Draganić et al. remains stronger since it imposes no lower bound on $\varepsilon$8. A notable claim is that the bipartite theorem appears to be the first Hamiltonicity result for bipartite expanders (linear or sublinear) with degree $\varepsilon$9.

Regularity is necessary: slightly unbalanced complete bipartite graphs are non-Hamiltonian expanders, and balanced non-Hamiltonian expanders exist even with all degrees within a factor $d \ge (\gamma^{-1}\varepsilon^{-1}\log n)^{10^8}$0 of each other. The paper therefore also establishes robust versions covering nearly regular expanders, where near-regularity must be much stronger than the expansion parameter, and, in the bipartite case, equal part sizes.

Proof architecture

The argument adapts the absorption method to sublinear expanders, following an outline inspired by Chakraborti, Janzer, Methuku and Montgomery's proof that dense graphs contain edge-disjoint cycles on the same vertex set. A reservoir $d \ge (\gamma^{-1}\varepsilon^{-1}\log n)^{10^8}$1, a $d \ge (\gamma^{-1}\varepsilon^{-1}\log n)^{10^8}$2-random vertex set with $d \ge (\gamma^{-1}\varepsilon^{-1}\log n)^{10^8}$3, serves as a connecting medium: adapting Bucić–Montgomery, $d \ge (\gamma^{-1}\varepsilon^{-1}\log n)^{10^8}$4 connects roughly $d \ge (\gamma^{-1}\varepsilon^{-1}\log n)^{10^8}$5 spread-out pairs by short internally disjoint paths. The remainder of the graph is decomposed into a linear forest via random partitioning and matching arguments (Vizing's theorem plus martingale concentration), whose endpoints are then linked through $d \ge (\gamma^{-1}\varepsilon^{-1}\log n)^{10^8}$6. Finally, unused reservoir vertices are absorbed via a dedicated absorber gadget.

The central technical novelty is a uniformity requirement: the absorber's vertex set must hit every neighbourhood in $d \ge (\gamma^{-1}\varepsilon^{-1}\log n)^{10^8}$7 vertices for $d \ge (\gamma^{-1}\varepsilon^{-1}\log n)^{10^8}$8 ($d \ge (\gamma^{-1}\varepsilon^{-1}\log n)^{10^8}$9-uniformity). Without this, deleting the absorber destroys near-regularity and the forest-decomposition fails. Achieving uniform absorbers requires the paper's key tool:

A random connecting lemma for sublinear expanders, likely of independent use. Given nearly regular alternating graphs mixing in time $n$0, and $n$1 prescribed endpoint pairs whose endpoint multisets are bi-uniform, it produces pairwise internally disjoint paths of odd length $n$2 such that the set of $n$3-th vertices across all paths is itself bi-uniform with parameter $n$4. The proof samples random walks of length $n$5 conditioned on endpoints, exploits rapid mixing (time $n$6, derived from spectral bounds including a Trevisan-based bound on $n$7 for far-from-bipartite expanders) so that path midpoints behave as uniform random vertices, discards intersecting paths iteratively (only a $n$8-fraction collide), and proves concentration of intersection counts via Talagrand's inequality together with a Poisson-paradigm argument establishing approximate independence between a fixed path containing a given vertex and its collision events. The bipartite variant replaces mixing time by bipartite mixing time and uses pair-absorbing gadgets indexed by a "robustly matchable" template multigraph of Montgomery, ensuring balanced consumption of both parts.

Three rounds of the connecting lemma assemble absorbers whose vertex sets inherit uniformity at parameters degrading from $n$9 through $G$0 down to $G$1, while remaining of size at most $G$2.

Applications

The robust theorems yield several consequences beyond Hamiltonicity per se.

Dense regular expanders: there exists $G$3 such that every $G$4-vertex $G$5-regular $G$6-expander with $G$7 and $G$8 is Hamiltonian unconditionally — no bipartiteness condition needed. Here graphs close to bipartite are handled by taking a maximum cut and repairing the imbalance and irregularity with a carefully constructed linear forest before invoking the bipartite robust theorem.

Robust Lovász conjecture for Cayley graphs: combining the robust theorems with a weak arithmetic regularity lemma of Bedert–Bucić–Kravitz–Montgomery–Müyesser, the paper shows that for a connected $G$9-vertex Cayley graph of degree $\gamma$0, the edge-percolated graph $\gamma$1 is Hamiltonian whp whenever $\gamma$2 for an absolute constant $\gamma$3. At $\gamma$4 this recovers the moderately sparse Cayley graph result of Bedert et al.; for linear-degree Cayley graphs it yields Hamiltonicity after percolation at $\gamma$5. Close-to-bipartite coset structures are handled using the Christofides–Hladký–Mathé iron-connectivity lemma, with expansion converted to iron-connectivity and vertex-transitivity exploited to show the auxiliary quotient graph is connected and vertex-transitive.

Kneser graphs: applying the spectral corollary to $\gamma$6 — an $\gamma$7-graph with $\gamma$8 — gives Hamiltonicity for $\gamma$9 above an absolute constant, complementing the structural breakthrough of Merino–Mütze–Namrata which settled Hamiltonicity of all Kneser graphs but without robustness. The robust version additionally implies Hamilton-connectedness (answering a question of Mütze in this range), $S \subseteq V(G)$0 edge-disjoint Hamilton cycles via iterative peeling, and whp Hamiltonicity of the edge-percolated Kneser graph.

Spanning subdivisions: under the far-from-bipartite hypotheses the graph contains a spanning subdivision of $S \subseteq V(G)$1 for any fixed $S \subseteq V(G)$2, and in the bipartite case a spanning subdivision of $S \subseteq V(G)$3 — strengthening the nearly-spanning results of Letzter–Methuku–Sudakov and complementing spanning clique subdivisions in strong linear expanders by Lee–Pavez-Signé–Petrov.

Limitations and open questions

The constants involved are large: the exponent $S \subseteq V(G)$4 on $S \subseteq V(G)$5 in the degree bound, the decomposition of the proof into three rounds of connecting with substantial slack in uniformity parameters, and the constant $S \subseteq V(G)$6 in the percolation threshold for Cayley graphs. More substantively, the methods fail for graphs that are close to bipartite but neither bipartite nor far from bipartite: random walks mix poorly across such structures, and the paper can only handle this regime at density $S \subseteq V(G)$7. Three explicit conjectures are stated: (i) polylogarithmic degree should suffice for arbitrary regular sublinear expanders without any bipartiteness condition; (ii) for bipartite $S \subseteq V(G)$8-regular $S \subseteq V(G)$9-expanders, degree polynomial in $1 \le |S| \le \tfrac{2}{3}n$0 (no logarithmic factor) should suffice; (iii) the analogous statement for $1 \le |S| \le \tfrac{2}{3}n$1-far-from-bipartite expanders. The authors also note that their random-walk approach fundamentally requires good edge expansion, and replacing it with vertex expansion would likely demand different techniques.

Conclusion

This work establishes Hamiltonicity of regular sublinear expanders at polylogarithmic degree under mild non-degeneracy conditions (bipartite or far from bipartite), supplies robust versions handling near-regularity, and introduces a random connecting lemma giving fine control over the pseudorandomness of connecting-path vertex sets. Its applications deliver robust strengthenings of state-of-the-art results on Cayley and Kneser graphs and extend the reach of the absorption method to the sublinear-expander setting, where previously only nearly-Hamilton cycles were known.