Colour-biased Hamilton cycles in randomly perturbed graphs

Wenchong Chen; Xinbu Cheng; Zhifei Yan

REVIEW 3 major objections 5 minor 23 references

Colour-biased Hamilton cycles in randomly perturbed graphs

T0 review · 3 major / 5 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read Adding O(n) random edges to any graph with minimum degree αn forces a Hamilton cycle with Ω(n) colour bias, and at the critical degree (r+1)n/2r the bias scales exactly with the number m of added random edges.

desk verdict Serious paper with a genuinely new critical-case phenomenon, but two load-bearing proof gaps (a false inequality in Thm 3.1 for r=2,3 and a failed degree-contradiction in Lemma 4.20) mean the main theorems are not proven as written. read the letter →

arxiv 2506.04189 v1 pith:FGJ25ZAC submitted 2025-06-04 math.CO

Wenchong Chen , Xinbu Cheng , Zhifei Yan This is my paper

classification math.CO MSC 05C4505C8005C1505C70

keywords colourbiasHamiltoncyclesrandomlyperturbedgraphsminimumdegreeedge-colouringsabsorptionmethodrandomdiscrepancy

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

A Hamilton cycle has t colour bias if it contains n/r + t edges of a single colour in an r-edge-colouring; the paper asks how many random edges must be added to a fixed graph of minimum degree αn to guarantee, no matter how the union is coloured, a Hamilton cycle whose colour distribution is forced away from perfect balance. The first result says that for every α>0 a linear number m=O(n) of random edges suffices for Ω(n) bias. The second, at the critical minimum degree (r+1)/2r, pins down the dependence on m: adding m random edges yields Ω(m) bias whenever m is superconstant, and there are colourings that keep every Hamilton cycle at O(m) bias, so the answer is Θ(m). This gives the randomly perturbed threshold for colour-biased Hamilton cycles and shows that the extremal balanced-colouring construction is the only obstruction.

What carries the argument

The proof of the critical case rests on a structural characterization (Theorem 4.2): if an r-coloured G_α with δ ≥ (r+1)n/2r has no Hamilton cycle with Ω(m) colour bias, then some induced subgraph on αn vertices is almost monochromatic—every matching avoiding the dominant colour c* has at most o(m) edges. Assuming this fails, the authors build a near-monochromatic cycle F of length Θ(m) whose vertices can be absorbed into a spanning cycle H in two ways, using 'bowties' (two triangles sharing a centre vertex) as switching devices that change the colour count by a controlled amount. Together with a Pósa-type spanning-cycle lemma and random-graph lemmas guaranteeing long almost-monochromatic paths and large matchings in every βn-set, this yields the Ω(m) bound.

What would settle it

Recompute the union bound in the proof of Lemma 4.20 with the paper's stated constants: two independent sets of size βn−$2^{{−4}}$β²m with intersection βn−$2^{{−6}}$β²m have union βn−7·$2^{{−6}}$β²m, which contradicts the claimed βn+$2^{{−6}}$β²m; checking whether a matching of size Ω(m) still exists under the weaker overlap is a direct finite calculation that would settle the lemma.

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Extended reading notes

Core claim

The paper establishes two complementary statements. For every α>0 and r≥2, there is m=O(n) such that G_α ∪ G(n,m) with δ(G_α)≥αn typically satisfies h_r(G_α ∪ G(n,m)) = Ω(n): no matter how the union's edges are r-coloured, a Hamilton cycle has many more than n/r edges of one colour. For 0<α<(r+1)/2r, m=o(n) random edges are useless for this purpose—there are graphs of that minimum degree whose every Hamilton cycle is perfectly balanced regardless of the added edges. The critical case α=(r+1)/2r is where the paper's main theorem lies: for any m with 1≪m≤n, the typical value of h_r is Ω(m), and matching constructions force O(m), so the bias is linear in the number of random edges.

Load-bearing premise

The lower bound at the critical degree relies on a lemma that every βn-sized vertex set either contains a large matching already or overlaps one of finitely many exceptional sets in a large independent set, with the overlap bounded away from the whole set; the paper's own inequalities do not force that overlap, so this inheritance step is the load-bearing assumption.

Editorial extensions

If this is right

For any positive minimum degree, a linear number of random edges makes colour bias as large as it can be, matching the deterministic dense threshold up to the constant factor.
At the critical minimum degree (r+1)/2r, even a superconstant number m of random edges produces a superconstant colour bias, so the transition from zero bias to linear bias is governed by how many random edges are present.
The upper-bound constructions show the Ω(m) lower bound is tight up to constants: there are colourings in which every Hamilton cycle has colour bias at most O(m).
The structural dichotomy—no large bias forces a large nearly monochromatic induced subgraph—identifies the balanced-colouring example as the unique extremal obstruction in the critical regime.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The structural dichotomy likely extends to a stability statement: graphs near the critical degree whose every Hamilton cycle is nearly balanced must contain a large nearly monochromatic set; a formal quantitative version could connect to discrepancy-stability results for other spanning structures.
The Θ(m) dependence suggests that the natural measure of 'randomness needed' for bias is the number of available random edges themselves; for m=n^{1/2}, the upper construction gives a concrete test case for algorithms that search for biased Hamilton cycles.
A testable extension is to push the Ω(m) result down to the regime where m is a constant (the current theorem needs 1≪m); a direct computation of the matching lemma at constant m would show whether the threshold is sharp at m=Θ(1).
The bowtie switching mechanism provides a colour-count identity that could be reused to bound the discrepancy of perfect matchings or small factors in randomly perturbed hypergraphs and graphs.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Two things to know about this paper. First, it targets the right question and identifies a genuinely new phenomenon: at the critical minimum degree (r+1)n/2r, the amount of colour bias in randomly perturbed graphs scales with the number m of added random edges, not with n. Second, the written proofs have two load-bearing gaps, and the paper as submitted does not prove its main theorems.

The good parts are real. The constructions in Section 2 are clean and correct: for alpha below critical, m=o(n) random edges can leave h_r=0, and at critical alpha there is an O(m) upper bound. The structural proposition (Theorem 4.2) is a sensible way to capture the obstruction to large bias, and the absorption framework for the perturbed setting is a natural combination of existing tools. If the gaps are patched, the results would be a solid contribution.

Now the gaps, in order of severity. In Theorem 3.1, the final inequality (2/(r+1)-5eps)n >= n/r+n/(2r) is simply false for r=2,3. For r=2 the left side is about 2n/3 and the right side is 3n/4; for r=3 it's n/2 - 5eps n vs n/2. So the claimed n/(2r) bias is not derived. The proof still yields Omega(n) bias for all r, but not the stated constant; the theorem as written is not established for r=2,3. This is arithmetically fixable if they weaken the bias constant, but as it stands it's wrong.

More serious is Lemma 4.20. The degree-contradiction step that is supposed to force (15) does not work: from (13) and failure of (15) you only get |I(U) union I(U_i)| >= beta n - 7*2^{-6} beta^2 m, not beta n + 2^{-6} beta^2 m. Consequently d(x) can be as large as (1-beta)n + O(beta^2 m), which is compatible with delta(G_alpha) >= alpha n when alpha = 1-beta. Since this lemma supplies the Omega(m) matching in W that drives the lower bound in Theorem 1.6, the critical-case lower bound is unsupported. The stress-test note is right; I checked the union bounds and the contradiction doesn't go through.

The concern about the greedy absorption in Lemma 3.6 looks like a minor issue, not a real gap: the matching edges used are vertex-disjoint, so the count of available edges after removals is fine.

Bottom line: this is a serious paper with the right idea, but the proof has two holes, one in the dense case and one in the critical case. It deserves a referee, not a desk reject, but the referee should be asked to check the constants and the Lemma 4.20 argument carefully. If the authors can fix those, the paper would likely be right. I would not cite it in its current form.

Referee Report

3 major / 5 minor

Summary. The paper studies colour-biased Hamilton cycles in randomly perturbed graphs, i.e. graphs of the form G_α ∪ G(n,m), where G_α has minimum degree at least α n. The main results claimed are: (1) for every α>0, adding O(n) random edges typically forces an Ω(n)-colour-biased Hamilton cycle in every r-edge-colouring; (2) below the threshold α < (r+1)/(2r), o(n) random edges do not suffice; (3) at the critical value α=(r+1)/(2r), adding m random edges typically forces an Ω(m)-colour-biased Hamilton cycle for every 1≪m≤n, and there are examples with O(m) bias. The proofs combine an absorption/sprinkling argument for the O(n)-edge regime and a structural decomposition of graphs without large colour bias for the critical regime, together with Pósa's theorem, Erdős-Gallai, and standard random graph estimates.

Significance. If the theorems are correct, the paper would settle the randomly perturbed threshold for colour-biased Hamilton cycles and give a sharp dependence on m at the critical endpoint, extending results of Freschi–Hyde–Lada–Treglown and Gishboliner–Krivelevich–Michaeli. The paper is written in a standard and readable style, and it contains explicit extremal constructions for the upper bounds. It also makes good use of existing tools without introducing fitted parameters. However, the correctness of the main lower-bound results currently rests on several proof gaps that need to be fixed before the claims can be accepted.

major comments (3)

[§4.3, Lemma 4.20] The proof of the overlap claim (15) is not valid. From (13) one has |I(U)|,|I(U_i)| ≥ βn − 2^{−4}β²m, and the negation of (15) gives |I(U)∩I(U_i)| < βn − 2^{−6}β²m. These imply |I(U)∪I(U_i)| ≥ 2(βn − 2^{−4}β²m) − (βn − 2^{−6}β²m) = βn − 7·2^{−6}β²m, which is smaller than βn, not larger. Therefore the displayed bound d_{G_α}(x) ≤ (1−β)n − 2^{−6}β²m does not follow, and the contradiction with δ(G_α) ≥ αn is not obtained. Since this step is used to guarantee that the random matching inside some U_i survives inside every U, and that matching is essential for the Ω(m) lower bound in Theorem 4.1 (and hence Theorem 1.6), this is a load-bearing gap. The argument may be repairable by choosing different constants and using the fact that an independent set in G_α has size at most βn, but a corrected proof must be supplied.
[§3.3, proof of Theorem 3.1] The final numerical claim in the proof is false. After equation (5), the paper asserts that the path P'_1 contains at least (2/(r+1) − 5ε)n − 2ε³n ≥ n/r + n/(2r) edges of one colour. But for every r ≥ 2 one has 2/(r+1) < (r+1)/(2r), so the coefficient 2/(r+1) is strictly smaller than the coefficient 3/(2r) = 1/r + 1/(2r) needed for the claimed n/(2r) colour bias. Thus the path P_1 obtained from the Gishboliner–Krivelevich–Michaeli lemma gives only Ω(n) bias of size about (r−1)n/(r(r+1)), not n/(2r). This invalidates the stated bound in Theorem 3.1 as written. Since Theorem 1.4 only requires Ω(n) bias, the proof may be salvageable by changing the target constant, but all the inequalities in this final step must be adjusted accordingly.
[§4.1, Lemma 4.5] The proof of Lemma 4.5 contains an unjustified averaging claim: it states that some colour c* has at least n·αn·r^{−1} > n²/(2r) edges. Averaging over the r colour classes gives only e_{c*} ≥ e(G)/r ≥ αn²/(2r) = (r+1)n²/(4r²), which is smaller than n²/(2r) for every r ≥ 2. As written, the Erdős–Gallai argument does not apply. This is repairable: since t = 2^{5r} b ≤ n/(32r), the Erdős–Gallai bound can be sharpened to ex(n,P_t) ≤ (t−2)n/2 ≤ n²/(64r), and the averaged colour count is larger than this. But the current proof needs this correction before the existence of the almost-monochromatic cycle F is established.

minor comments (5)

[§2] The numbering in Section 2 is inconsistent: Lemma 2.1 is proved under the name “Proof of Theorem 2.1”, and Lemma 2.2 is referred to as “Theorem 2.2”.
[§2, Proposition 1.5] Proposition 1.5 in the introduction is called “Theorem 1.5” in Section 2; the labels should be harmonized.
[§4.2, Lemma 4.19] In the statement of Lemma 4.19, the phrase “at least t more edges of colour c” should refer to ℓ, the number of bowties, rather than t; as written it is ambiguous because t is defined earlier as 2^{5r} b.
[§4.2, Lemma 4.16] In the proof of Lemma 4.16, when the selected side-edge colour c' equals c, the cycle H_2 has one more c-edge than H_1, not H_1 more than H_2. The conclusion still follows if the argument is phrased as a difference of ℓ/r in absolute value, but the sentence as written is not correct in that case.
[§3.2, proof of Lemma 3.6] In the greedy connection step, the proof says there are at least 4εn choices and that at most 4εn choices need to be forbidden; this only guarantees a surviving choice if the inequalities are strict or the constants are adjusted. A small constant slack would make the argument clean.

Circularity Check

0 steps flagged · score 0.0 of 10

No circular reasoning found; the derivation chain uses external theorems and stated assumptions, with no fitted parameters or self-citation carrying the argument.

full rationale

The paper derives its theorems by standard external tools (Pósa's theorem, Erdős–Gallai, Kim–Vu concentration, Chernoff bounds, and results from GKM22, KLS15, and FHLT21) applied to explicitly stated graph classes. The main results are proved from the assumptions δ(G)≥αn and the definition of G(n,m), without fitting parameters to target quantities or defining the desired conclusion into the hypotheses. For instance, Theorem 1.4 is assembled from Theorem 3.2 (an external almost-monochromatic path lemma), Theorem 3.6 (an absorber constructed via explicit matching arguments), and Theorem 3.3 (a path in βn-subsets derived from the k-joined property), none of which assumes the biased Hamilton cycle being proved. Similarly, Theorem 4.1 is built from the structural Proposition 4.2, the bowtie lemmas, and Lemma 4.20; the proof of Lemma 4.20 does contain an apparent gap in the inequality (15) (the union bound derivation is not valid as written), but that is a correctness issue, not a circularity: the proof still attempts to derive the matching property from δ(G_α)≥αn and random-edge concentration, and it does not assume the conclusion. The upper-bound constructions in Section 2 are explicit and match the claimed bounds through direct counting. The acknowledgements and references include self-citations only in the ordinary sense of building on prior work, and no central premise is justified solely by an unpublished or same-author uniqueness claim. Overall, the analysis finds no step where a 'prediction' or theorem statement is equivalent by construction to its inputs.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The central claims rest entirely on standard external theorems and probabilistic inequalities. There are no fitted free parameters and no new postulated objects. The main risk identified in this review is incorrect derivation from these tools, not unsupported axioms.

assumptions (5)

standard math Pósa's theorem: a graph on n vertices with δ≥(n+ℓ)/2 contains a Hamilton cycle containing any given path forest with ℓ edges.
Invoked as Theorem 4.6 in Section 4 to find cycles through prescribed matchings and path forests.
standard math Erdős-Gallai bound ex(n,P_t) ≤ (t−2)n/2.
Used in Lemma 4.5 to guarantee a long monochromatic path in a colour class.
standard math GKM22 DFS lemma: G(n,p) with p≥C/n has an almost monochromatic path of length (2/(r+1)−ε)n in any r-colouring.
Quoted as Lemma 3.2; this external result supplies the long monochromatic path in Section 3.
standard math Kim-Vu inequality and Chernoff bounds for random graph concentration.
Used in the appendix to control e(R), Y(R), δn-joinedness, and matching sizes.
standard math Janson-Luczak-Ruciński monotone conversion between G(n,p) and G(n,m).
Used at the end of Theorem 4.1 to pass from the p-model to the m-model.

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Cite this review

Pith. "Pith review of Colour-biased Hamilton cycles in randomly perturbed graphs." pith.science (2026). https://pith.science/paper/FGJ25ZAC

@misc{pith2026250604189,
  author       = {Pith},
  title        = {Pith review of: Colour-biased Hamilton cycles in randomly perturbed graphs},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/FGJ25ZAC}},
  note         = {Machine review of arXiv:2506.04189}
}

abstract

Given a graph $G$ and an $r$-edge-colouring $\chi$ on $E(G)$, a Hamilton cycle $H\subset G$ is said to have $t$ colour-bias if $H$ contains $n/r+t$ edges of the same colour in $\chi$. Freschi, Hyde, Lada and Treglown showed every $r$-coloured graph $G$ on $n$ vertices with $\delta(G)\geq(r+1)n/2r+t$ contains a Hamilton cycle $H$ with $\Omega(t)$ colour-bias, generalizing a result of Balogh, Csaba, Jing and Pluh\'{a}r. In 2022, Gishboliner, Krivelevich and Michaeli proved that the random graph $G(n,m)$ with $m\geq(1/2+\varepsilon)n\log n$ typically admits an $\Omega(n)$ colour biased Hamilton cycle in any $r$-colouring. In this paper, we investigate colour-biased Hamilton cycles in randomly perturbed graphs. We show that for every $\alpha>0$, adding $m=O(n)$ random edges to a graph $G_\alpha$ with $\delta(G_\alpha)\geq \alpha n$ typically ensures a Hamilton cycle with $\Omega(n)$ colour bias in any $r$-colouring of $G_\alpha\cup G(n,m)$. Conversely, for certain $G_{\alpha}$, reducing the number of random edges to $m=o(n)$ may eliminate all colour biased Hamilton cycles of $G(n,m)\cup G$ in a certain colouring. In contrast, at the critical endpoint $\alpha=(r+1)/2r$, adding $m$ random edges typically results in a Hamilton cycle with $\Omega(m)$ colour-bias for any $1\ll m\leq n$.

Figures

Figures reproduced from arXiv: 2506.04189 by the authors.

**Figure 2.** A cycle containing the side-edges of bowties First, if there exists a colour c ∈ [r] such that |Xc| ⩾ s/4, we can find a collection of vertex-disjoint bowties of c-type I containing the vertices in |Xc|, defined as follows. Definition 4.14. A bowtie B is of c-type I for c ∈ [r], if it has 4 center-edges of colour c, and 2 side-edges whose colours are distinct from each other. Note that for each v ∈ Xc and each colou… view at source ↗

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Reference graph

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Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.

[1] [1]

Minimum Degree Threshold for $H$-factors with High Discrepancy

D. Brada c , M. Christoph, and L. Gishboliner. Minimum degree threshold for h -factors with high discrepancy. arXiv:2302.13780

work page Pith review arXiv

[2] [2]

Balogh, B

J. Balogh, B. Csaba, Y. Jing, and A. Pluh\' a r. On the discrepancies of graphs. Electron. J. Combin. , 27(2):Paper No. 2.12, 14, 2020

work page 2020

[3] [3]

Bennett, A

P. Bennett, A. Dudek, and A. Frieze. Adding random edges to create the square of a H amilton cycle. arXiv:1710.02716

arXiv

[4] [4]

Bohman, A

T. Bohman, A. Frieze, and R. Martin. How many random edges make a dense graph H amiltonian? Random Structures & Algorithms , 22(1):33--42, 2003

work page 2003

[5] [5]

B \"o ttcher, R

J. B \"o ttcher, R. Montgomery, O. Parczyk, and Y. Person. Embedding spanning bounded degree graphs in randomly perturbed graphs. Mathematika , 66(2):422--447, 2020

work page 2020

[6] [6]

Balogh, A

J. Balogh, A. Treglown, and A. Wagner. Tilings in randomly perturbed dense graphs. Combinatorics, Probability and Computing , 28:159--176, 2019

work page 2019

[7] [7]

Das and A

S. Das and A. Treglown. Ramsey properties of randomly perturbed graphs: cliques and cycles. Combinatorics, Probability and Computing, to appear , 2020

work page 2020

[8] [8]

Freschi, J

A. Freschi, J. Hyde, J. Lada, and A. Treglown. A note on color-bias H amilton cycles in dense graphs. SIAM J. Discrete Math. , 35(2):970--975, 2021

work page 2021