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There exists an absolute constant K such that any n-vertex d-regular γ-expander with d ≥ (γ^{-1} log n)^K is Hamiltonian if bipartite or γ-far from bipartite.

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load-bearing objection The paper gives a Hamiltonicity threshold for d-regular γ-expanders at d ≥ (γ^{-1} log n)^K when the graph is bipartite or γ-far from bipartite, plus a random connecting lemma for sublinear expanders. the 2 major comments →

arxiv 2605.15043 v1 pith:GJYSDDLV submitted 2026-05-14 math.CO

Hamiltonicity of regular sublinear expanders

Domagoj Brada\v{c} , Oliver Janzer This is my paper
classification math.CO
keywords Hamiltonicityexpandersregular graphsbipartite graphsCayley graphsKneser graphsconnecting lemma
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper establishes a degree threshold guaranteeing that regular expander graphs contain a cycle through every vertex. For graphs meeting the expansion condition and either bipartite or sufficiently far from bipartite, a degree polynomial in the inverse of the expansion parameter and the logarithm of n suffices. The proof introduces a random connecting lemma for sublinear expanders to construct the necessary paths. This yields a general criterion that applies uniformly rather than to isolated families of graphs.

Core claim

We prove that there exists an absolute constant K such that any n-vertex d-regular γ-expander with d ≥ (γ^{-1} log n)^K is Hamiltonian, provided that it is bipartite or γ-far from bipartite. As part of the proof we establish a random connecting lemma for sublinear expanders.

What carries the argument

The random connecting lemma for sublinear expanders, which produces paths between prescribed vertices while preserving the expansion properties needed to complete a Hamiltonian cycle.

Load-bearing premise

The random connecting lemma for sublinear expanders holds and can be applied to produce the required paths and cycles under the stated degree threshold.

What would settle it

An n-vertex d-regular γ-expander with d just below (γ^{-1} log n)^K that is bipartite yet contains no Hamiltonian cycle would falsify the claim.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • Highly robust versions of recent Hamiltonicity results for Cayley graphs follow directly from the degree threshold.
  • Highly robust versions of recent Hamiltonicity results for Kneser graphs follow directly from the degree threshold.
  • The random connecting lemma supplies a tool for constructing paths in other sparse expander settings.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same degree threshold supplies a uniform test that can be applied to additional families of regular graphs once their expansion parameters are known.
  • The connecting lemma may extend to questions of long paths or cycles in sublinear expanders that are not necessarily regular.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

2 major / 2 minor

Summary. The paper proves that there exists an absolute constant K such that any n-vertex d-regular γ-expander with d ≥ (γ^{-1} log n)^K is Hamiltonian, provided the graph is bipartite or γ-far from bipartite. The proof introduces a random connecting lemma for sublinear expanders, which is used to construct the required paths and cycles, and applies the main theorem to obtain robust Hamiltonicity results for Cayley graphs and Kneser graphs.

Significance. If the random connecting lemma and its application to the degree threshold hold, the result substantially strengthens the known Hamiltonicity thresholds for regular expanders, moving from linear to polylogarithmic degree in the expansion parameter. The applications yield highly robust versions of recent results on Cayley and Kneser graphs, and the connecting lemma may have independent value for other connectivity problems in sparse expanders.

major comments (2)
  1. [§3] §3 (Random Connecting Lemma): the quantitative bound on the number of random walks needed to connect two sets appears to rely on a Chernoff-type tail bound whose failure probability must be union-bounded over all pairs of sets; it is not immediately clear from the statement whether the resulting degree threshold (γ^{-1} log n)^K absorbs all logarithmic factors arising from this union bound.
  2. [Theorem 1.1] Theorem 1.1 and the bipartite case: the argument that the lemma produces a Hamilton cycle when the graph is bipartite uses a perfect matching between the two parts; the proof sketch does not explicitly verify that the expansion parameter γ remains sufficient after removing the matching edges, which could affect the sublinear regime.
minor comments (2)
  1. [Abstract / Theorem 1.1] The definition of 'γ-far from bipartite' (at least γ e(G) edges to remove) is used in the non-bipartite case but is not restated in the statement of the main theorem; adding a parenthetical reminder would improve readability.
  2. [Introduction] Notation for the expansion parameter γ is overloaded between the expander definition and the far-from-bipartite condition; a brief remark distinguishing the two uses would prevent confusion.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the positive assessment and constructive comments. Below we respond point-by-point to the major comments; all clarifications will be incorporated in the revised manuscript.

read point-by-point responses

  1. Referee: [§3] §3 (Random Connecting Lemma): the quantitative bound on the number of random walks needed to connect two sets appears to rely on a Chernoff-type tail bound whose failure probability must be union-bounded over all pairs of sets; it is not immediately clear from the statement whether the resulting degree threshold (γ^{-1} log n)^K absorbs all logarithmic factors arising from this union bound.

    Authors: In the proof of the Random Connecting Lemma, the number of walks per pair is chosen on the order of (log n / γ)^C for a sufficiently large C so that the Chernoff tail is exp(−Ω((log n / γ)^c)) for some c>0. The union bound is over at most n^2 pairs of sets and contributes an additional O(log n) to the exponent. The absolute constant K in the degree hypothesis d ≥ (γ^{-1} log n)^K is chosen larger than all constants appearing in these calculations (including those from the Chernoff and union-bound exponents), so that the failure probability is o(1/n^2) and the threshold absorbs every logarithmic factor. We will add an explicit remark after the statement of the lemma recording the dependence of K on these quantities. revision: yes

  2. Referee: [Theorem 1.1] Theorem 1.1 and the bipartite case: the argument that the lemma produces a Hamilton cycle when the graph is bipartite uses a perfect matching between the two parts; the proof sketch does not explicitly verify that the expansion parameter γ remains sufficient after removing the matching edges, which could affect the sublinear regime.

    Authors: In the bipartite case we delete a perfect matching M and work in G−M, which is (d−1)-regular. For any set S the number of edges leaving S in G−M is at least γ d |S| − |S|, hence the new expansion parameter satisfies γ′ ≥ (γ d − 1)/(d − 1). Whenever d ≥ 2/γ we have γ′ ≥ γ/2. Our degree lower bound already guarantees d ≫ 1/γ (since K ≥ 1), so the halved parameter γ/2 only multiplies the threshold by a constant factor 2^K, which is absorbed into the choice of the absolute constant K. We will insert this short calculation immediately after the invocation of the connecting lemma in the bipartite case of Theorem 1.1. revision: yes

Circularity Check

0 steps flagged

Derivation is self-contained; no circular reductions identified

full rationale

The paper establishes its Hamiltonicity result by proving a new random connecting lemma for sublinear expanders, presented as an independent contribution. The abstract and description contain no self-definitional steps, fitted inputs renamed as predictions, or load-bearing self-citations that reduce the central claim to its own inputs by construction. The bipartite and γ-far-from-bipartite cases are handled directly within the proof structure, and applications to Cayley/Kneser graphs are straightforward consequences once the lemma holds. No equations or lemmas are quoted that exhibit the forbidden patterns of circularity.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Central claim rests on the standard definitions of γ-expander and γ-far from bipartite plus the new random connecting lemma; no free parameters or invented entities are visible in the abstract.

axioms (2)
  • domain assumption Definition of γ-expander: for every not too large S there are at least γ d |S| edges leaving S.
    Standard definition invoked in the statement of the main theorem.
  • domain assumption Definition of γ-far from bipartite: at least γ e(G) edges must be removed to make G bipartite.
    Standard definition invoked in the statement of the main theorem.

pith-pipeline@v0.9.1-grok · 5684 in / 1227 out tokens · 29621 ms · 2026-06-30T19:59:26.785340+00:00 · methodology

0 comments
read the original abstract

We say that a $d$-regular graph is a $\gamma$-expander if for every not too large set of vertices $S$, there are at least $\gamma d |S|$ edges leaving $S$, and we say that a graph $G$ is $\gamma$-far from bipartite if at least $\gamma e(G)$ 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 $\gamma$-expander with $d \ge (\gamma^{-1} \log n)^K$ is Hamiltonian, provided that it is bipartite or $\gamma$-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.

Figures

Figures reproduced from arXiv: 2605.15043 by Domagoj Brada\v{c}, Oliver Janzer.

Figure 1
Figure 1. Figure 1: The construction of an (x, a, b, y)-absorber with ℓ = 7. The blue segments and curves represent single edges, where the a − b path with arcs below corresponds to Q1 and the a − b path with arcs above corresponds to Q2. The red segments represent paths P1, . . . , Pℓ. The first path is an x−y path with vertex set V (H) while the second is an x−y path with vertex set V (H)\ {a, b}. Finally, we use the follow… view at source ↗
Figure 2
Figure 2. Figure 2: The construction of an (x, a, y)-absorber with ℓ = 7. The blue segments and arcs represent single edges, where the a − y path with arcs below corresponds to Q1 and the a − u6 path with arcs above to Q2. The red segments represent paths P1, . . . , Pℓ−1. Construction of an (x, a, y)-absorber. Suppose we are given distinct vertices x, a, y in a graph G and an odd integer ℓ. Let Q1 = u0, . . . , uℓ be an a − … view at source ↗
Figure 3
Figure 3. Figure 3: An example with m = 3. The red, blue and green paths are constructed using the connecting properties of R. The orange lens-shaped objects represent (x, a, b, y)-absorbers. 29 [PITH_FULL_IMAGE:figures/full_fig_p029_3.png] view at source ↗

discussion (0)

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Forward citations

Cited by 1 Pith paper

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  1. Towards the Lov\'{a}sz conjecture via sublinear expanders

    math.CO 2026-06 unverdicted novelty 7.0

    Every connected vertex-transitive graph of order n contains a cycle of length at least n^{2/3-o(1)}.

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