Small circumference in regular sublinear expanders

Abstract: Sublinear expansion is weak enough to be extracted from arbitrary graphs while retaining nearly all of their average degree, yet it has proved strong enough to force global structures in many sparse extremal problems. Letzter, Methuku and Sudakov [JLMS 2026] developed methods yielding nearly Hamilton cycles in sufficiently dense regular sublinear expanders, and Montgomery [ICM 2026] subsequently conjectured that, every sufficiently large (but constant) degree $d$-regular sublinear expander is Hamiltonian. We disprove this conjecture in a strong form by constructing $n$-vertex $d$-regular sublinear expanders with degree $d=\left(\frac12+o(1)\right)\log² n$, which does not even has a cycle covering a positive fraction of its vertices. The construction blows up one side of a biregular Ramanujan graph into almost-complete blocks while keeping the other side independent. The Ramanujan incidence graph certifies expansion for arbitrary mixtures of partial blocks and separator vertices, whereas the independent side forms a sparse vertex separator that prevents a cycle from visiting enough blocks. The construction also explains why $\log² n$ is the natural degree scale for this obstruction.

• The paper disproves Montgomery’s Hamiltonicity conjecture by constructing infinitely many exactly d-regular sublinear expanders with d=(1/2+o(1))log²n and circumference below ηn for every fixed η<1/2.
• The construction combines near-complete blocks with an independent separator connected through a biregular Ramanujan graph, preserving expansion while limiting every cycle to fewer than ηn vertices.
• The block–separator method shows that the obstruction naturally operates at the log²n degree scale, leaving open whether every sufficiently large regular (ε,d)-expander with d≥C log²n must eventually be Hamiltonian.

Overview and main result

This paper by Chen, Liu, Wei, and Yang disproves a conjecture of Montgomery on Hamiltonicity of regular sublinear expanders, in a substantially stronger form than mere non-Hamiltonicity. Montgomery conjectured that for every $\varepsilon > 0$ there exists $d_0$ such that every $d$-regular $(\varepsilon, d)$-expander with $d \ge d_0$ contains a Hamilton cycle (Montgomery, 28 Jul 2026). The main theorem refutes this: for every fixed $0 < \eta < 1/2$ there exist infinitely many pairs $(d, n)$ and $n$-vertex $d$-regular graphs that are $(\varepsilon_1, \varepsilon_2 d)$-expanders with

$d_0$0

yet have circumference $d_0$1. In other words, no cycle covers even a positive fraction of the vertices — a much stronger failure than absence of a Hamilton cycle. Since the constructed graphs are also $d_0$2-expanders under Montgomery's normalization (the authors verify this via monotonicity of $d_0$3 in $d_0$4), the counterexamples apply directly to the original conjecture.

The result stands in sharp contrast to two prior lines of work: Letzter, Methuku and Sudakov proved that $d_0$5-vertex $d_0$6-regular sublinear expanders with $d_0$7 contain cycles of length at least $d_0$8 [2608-related JLMS result], and Draganić, Montgomery, Munhá Correia, Pokrovskiy and Sudakov resolved the analogous Krivelevich–Sudakov spectral conjecture affirmatively (Draganić et al., 2024). Bradač and Janzer showed Hamiltonicity of regular edge expanders above degree $d_0$9, with additional bipartiteness hypotheses (Bradač et al., 14 May 2026). The present paper demonstrates that exact regularity alone does not rescue spanning behavior at moderate polylogarithmic degrees.

Construction

The graph $d$0 is built from an auxiliary $d$1-biregular bipartite Ramanujan graph $d$2 with bipartition $d$3, where $d$4 and $d$5. Each vertex of $d$6 is blown up into a block $d$7, where $d$8 is a matching of size $d$9; the $(\varepsilon, d)$0 endpoints of $(\varepsilon, d)$1 are the ports, identified bijectively with the edges of $(\varepsilon, d)$2 incident to vertex $(\varepsilon, d)$3. Every edge $(\varepsilon, d)$4 is realized as an edge from $(\varepsilon, d)$5 to the corresponding port, and $(\varepsilon, d)$6 is kept as an independent set.

Two structural features drive the entire argument:

• Dense blocks: each block is nearly complete, so internal expansion within touched blocks is essentially automatic, and ports (degree $(\varepsilon, d)$7 internally plus one edge to $(\varepsilon, d)$8) recover regularity.
• Independent separator: since $(\varepsilon, d)$9 is independent and $d \ge d_0$0 is a disjoint union of blocks, deleting the vertices a cycle uses in $d \ge d_0$1 leaves paths each confined to a single block.

Existence of $d \ge d_0$2 follows from the Marcus–Spielman–Srivastava interlacing-families machinery applied to iterated 2-lifts of $d \ge d_0$3 [MSS], yielding the Ramanujan bound $d \ge d_0$4 together with the exact biregularity and part sizes required ($d \ge d_0$5, $d \ge d_0$6). The parameter $d \ge d_0$7 is chosen maximally so that $d \ge d_0$8, which forces the relation $d \ge d_0$9. With $0 < \eta < 1/2$0, so $0 < \eta < 1/2$1, the resulting graph is simple and exactly $0 < \eta < 1/2$2-regular on $0 < \eta < 1/2$3 vertices.

Circumference obstruction

Let $0 < \eta < 1/2$4 be any cycle and set $0 < \eta < 1/2$5. If $0 < \eta < 1/2$6, then $0 < \eta < 1/2$7 lies inside one block and has length at most $0 < \eta < 1/2$8. Otherwise, deleting $0 < \eta < 1/2$9 splits $(d, n)$0 into $(d, n)$1 vertex-disjoint paths, each contained in one block; hence $(d, n)$2 meets at most $(d, n)$3 blocks. Since each block contributes at most $(d, n)$4 vertices to the cycle,

$(d, n)$5

for all sufficiently large $(d, n)$6. This is a clean counting obstruction: the sparse separator caps how many dense pieces any cycle can stitch together, and it immediately explains why the failure is quantitative rather than marginal.

Verification of sublinear expansion

The technical heart of the paper is showing that $(d, n)$7 is an $(d, n)$8-expander despite the separator. For $(d, n)$9 with $n$0, let $n$1 index the blocks met by $n$2, let $n$3, let $n$4 be the number of singleton blocks among those indexed by $n$5, and let $n$6 count omitted vertices in touched blocks. A key observation is that because each vertex of $n$7 has at most one non-neighbor inside its own block, every non-port vertex of a touched block with $n$8 lies in the neighborhood; only singleton blocks lose internal expansion, contributing the term $n$9 from internal neighbors.

The external neighborhood decomposes into three disjoint parts $d$0, $d$1, $d$2: internal neighbors in touched blocks, ports reached from $d$3 into untouched blocks (counted exactly by $d$4), and separator vertices exposed by touched blocks. The interface quantity

$d$5

is bounded below using the expander mixing lemma and a spectral neighborhood estimate derived from it: every $d$6 satisfies $d$7 where $d$8. Combining both directions of the interface gives $d$9 with $(\varepsilon_1, \varepsilon_2 d)$0.

The decomposition loses at most $(\varepsilon_1, \varepsilon_2 d)$1 (ports already counted as internal neighbors), while the $(\varepsilon_1, \varepsilon_2 d)$2 singleton blocks contribute disjoint internal neighborhoods of size at least $(\varepsilon_1, \varepsilon_2 d)$3. Taking the maximum of these two estimates yields

$(\varepsilon_1, \varepsilon_2 d)$4

which is the central reconciliation step: the loss from singletons in the interface bound is exactly compensated by their large internal neighborhoods. When $(\varepsilon_1, \varepsilon_2 d)$5, a direct count gives $(\varepsilon_1, \varepsilon_2 d)$6. Finally, the choice of $(\varepsilon_1, \varepsilon_2 d)$7 ensures that even the maximal requirement $(\varepsilon_1, \varepsilon_2 d)$8, reducing the verification to the linear-scale inequality $(\varepsilon_1, \varepsilon_2 d)$9; when instead $d_0$00, the bound $d_0$01 dominates $d_0$02 since $d_0$03 for $d_0$04.

Why the logarithmic-square scale is intrinsic

The construction admits no substantial increase in degree. Taking $d_0$05 to be a union of $d_0$06 whole blocks gives $d_0$07 but $d_0$08 with $d_0$09; sublinear expansion at this scale demands a neighborhood of order $d_0$10, forcing $d_0$11. The exponent $d_0$12 is therefore inherent to the block–separator mechanism, though the authors are careful not to claim it is inherent to the Hamiltonicity problem itself. Similarly, the dependence of $d_0$13 on $d_0$14 is unavoidable in this scheme: a whole block requires $d_0$15, while the circumference bound needs $d_0$16, so $d_0$17 cannot tend to zero at fixed $d_0$18.

Limitations and open questions

The paper concedes several points explicitly. First, the counterexamples live at $d_0$19; they do not rule out Hamiltonicity at smaller degrees or rule out that some positive-side theorem holds already at this scale. Second, all conclusions are specific to this one-level block–separator architecture — the impossibility statements about the degree scale and the $d_0$20–$d_0$21 tradeoff apply to the construction, not to arbitrary potential obstructions. Third, the expansion verification relies on the Ramanujan spectral guarantee and the near-completeness of the blocks; whether sparser blocks could yield similar counterexamples is not addressed.

The principal question left open is stated precisely: does there exist $d_0$22 such that every sufficiently large $d_0$23-vertex $d_0$24-regular $d_0$25-expander with $d_0$26 contains a Hamilton cycle? The gap between the counterexamples at $d_0$27 and the nearly-Hamilton threshold at $d_0$28 is now the natural target.

Conclusion

This paper settles Montgomery's conjecture negatively and quantitatively: exact regularity of a sublinear expander does not force even linearly long cycles once the degree reaches the logarithmic-square scale. The construction couples a biregular Ramanujan incidence graph — supplying spectral control over arbitrary mixtures of partial blocks and separator vertices — with almost-complete blocks and an independent sparse separator that caps the circumference below $d_0$29. Beyond refuting the conjecture, the work identifies $d_0$30 as the intrinsic scale of this obstruction and isolates the precise open problem of whether that same scale marks the true Hamiltonicity threshold for regular sublinear expanders.