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Symmetry can't save directed cycles from missing 1 in 12 vertices

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2026-07-08 23:26 UTC pith:DYYDN657

load-bearing objection Two strong results: a clean parity obstruction resolving a conjecture, and a weighted cycle graph framework improving the directed cycle lower bound to match Babai. Proofs check out under careful reading. the 3 major comments →

arxiv 2607.05807 v1 pith:DYYDN657 submitted 2026-07-07 math.CO

Long Directed Cycles in Vertex-Transitive Digraphs

Bowen Li , Abhishek Methuku This is my paper
classification math.CO MSC 05C2005C4505C38
keywords vertex-transitive digraphdirected cycleHamiltonian cycleperimeter gapparity obstructioncycle graphweighted diameter
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.

This paper proves two results about directed cycles in vertex-transitive digraphs—graphs that look the same from every vertex. First, it constructs infinitely many such digraphs on n vertices whose longest directed cycle misses at least n/12 vertices, confirming a conjecture that the gap can grow linearly. The construction uses a layered graph where each layer is a 12-element set, and a parity argument shows that any Hamiltonian cycle would force an impossible composition of even permutations into an odd one. Second, it proves that every connected vertex-transitive digraph on n vertices contains a directed cycle of length at least proportional to the square root of n, matching the classical undirected bound. The key innovation is a weighted cycle graph that weights each vertex by the length of the corresponding directed cycle, eliminating a multiplicative loss that limited prior approaches to a cube-root bound.

Core claim

The central mechanism is a parity obstruction in a layered vertex-transitive digraph. Each layer is a copy of a 12-element set X of ordered pairs from {1,2,3,4}. Edges go from layer t to layer t+1, mapping (a,b) to (b,c) with c distinct from both. Any bijection between consecutive layers induced by a Hamiltonian cycle decomposes into four disjoint 3-cycles composed with six transpositions, yielding an even permutation. Composing m such even permutations around the full graph gives an even first-return permutation on a layer, but a Hamiltonian cycle requires this permutation to be a single 12-cycle, which is odd—a contradiction. For the lower bound, the central object is a weighted cyclegraph

What carries the argument

Weighted cycle graph C(D) where each vertex (a directed cycle of D) has weight equal to its length; weighted diameter controls directed diameter; near-transitivity of C(D) inherited from vertex-transitivity of D; induced cycles in C(D) admit low-weight shortcuts via a hub vertex of weight at most the circumference.

Load-bearing premise

The lower bound proof depends on a structural folding argument (Lemma 3.8) showing that a large weighted diameter forces an induced cycle with a heavy geodesic subpath; this argument involves multiple overlapping path constructions where subtle errors in weight accounting could propagate, and the unoptimized constant 300 reflects the complexity of this step.

What would settle it

Construct a connected vertex-transitive digraph on n vertices where all directed cycles have length o(sqrt(n)), which would contradict Theorem 1.5.

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

If this is right

  • The parity obstruction technique may generalize to other layer sizes beyond 12, potentially yielding larger constant-factor perimeter gaps or applying to other symmetry classes.
  • The weighted cycle graph framework could improve long-cycle bounds for other structured digraph families where an auxiliary intersection graph is available.
  • The gap between the n/12 construction and the square-root-of-n guarantee leaves a wide range open; the true extremal behavior remains unknown.
  • The linear diameter-circumference relation may find use in other problems where path-to-cycle extraction incurs metric loss.

Where Pith is reading between the lines

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

  • If the constant 300 in the diameter-circumference relation could be reduced, the square-root bound would improve, but the construction showing n/12 gap suggests the square-root order may be tight up to constants.
  • The parity obstruction is fundamentally different from prior number-theoretic obstructions and may apply to Cayley digraphs where arithmetic constraints are absent.
  • Whether the n/12 constant can be improved toward n/2 or beyond is a natural question, since the maximum possible perimeter gap for a connected vertex-transitive digraph is n minus the shortest cycle length.

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

3 major / 6 minor

Summary. This paper makes two contributions to the theory of long directed cycles in vertex-transitive digraphs. First, it resolves a conjecture of Bucić, Hendrey, Mohar, Steiner, and Yepremyan by constructing, for infinitely many n, a connected vertex-transitive digraph on n vertices with perimeter gap at least n/12. The construction uses a cyclic layered digraph on X × Z_m (where |X| = 12) and a combinatorial parity obstruction to rule out Hamiltonicity. Second, it improves the best known lower bound on the longest directed cycle in every connected vertex-transitive digraph from Ω(n^{1/3}) to Ω(√n), matching Babai's classical undirected bound. The proof introduces a weighted cycle graph C(D) with weight w(F) = |F| and establishes a linear relation ℓ ≥ d/300 between the circumference ℓ and the directed diameter d.

Significance. Both results represent substantial advances. Theorem 1.3 resolves Conjecture 1.2 of [4] in a strong form, upgrading the logarithmic perimeter gap to a linear one. The parity obstruction is conceptually clean and self-contained. Theorem 1.5 improves the Ω(n^{1/3}) bound of [4] to Ω(√n), which is the natural barrier given Babai's undirected result. The weighted cycle graph machinery (Lemmas 3.6–3.8) is the key technical innovation: by weighting each cycle vertex by its length, the authors eliminate the multiplicative loss that limited the unweighted approach. The constant 300 is explicitly unoptimized, which is appropriate. The construction in Section 4 is fully explicit and verifiable.

major comments (3)
  1. Lemma 3.7, construction of Z: The proof argues that the concatenation of subpaths of F_1,…,F_t forms a simple directed cycle Z. The key subtlety is that F_i ∩ F_{i+1} may contain multiple vertices, and the 'first vertex encountered' choice for v_{i+1} along F_i must ensure that the subpath of F_i from v_i to v_{i+1} does not internally meet F_{i+1}. The paper states this correctly but tersely: 'the choice of v_{i+1} as the first vertex of F_{i+1} encountered along F_i ensures that the corresponding subpath of F_i is internally disjoint from F_{i+1}.' This is correct, but the argument also needs that the subpaths are pairwise internally disjoint across non-adjacent indices (which follows from H being induced, t ≥ 4). The authors should add one or two sentences making the pairwise disjointness argument fully explicit, as it is load-bearing for the claim that Z is a simple cycle and hence |
  2. Lemma 3.8, Claim 3.9, inequality chain (2)–(6): The bound min{A,B} ≤ 7W is the crux of the folding argument. The chain combines the geodesicity of S' and Q with the fact that each edge in G has weight at most 2W (since W = max w(v)). The step from (5) to the final bound uses |A − D_a| ≤ 4W and |B − D_b| ≤ 4W, which account for at most two adjacency steps (a to a', u to u'), each contributing at most 2W. This is correct. However, the constant 7W arises as 14W/2, and the 14W itself comes from 5W + 4W + 4W + W. The authors should verify that the W in (2) (subtracting w(u')) and the 4W bounds are not double-counting the same vertex weight. As written, the accounting appears correct, but a brief remark clarifying that the −W in (2) and the +4W bounds refer to distinct quantities would strengthen the argument.
  3. Lemma 3.8, Claim 3.10, equation (11): The weight comparison w(T) ≥ w(P) − 21W + (Γ/2 − 11W) requires Γ ≥ 300W to yield w(T) > w(P). Specifically, the bound gives w(T) − w(P) ≥ Γ/2 − 32W, which is positive when Γ > 64W. The assumption Γ ≥ 300W provides ample margin. This is correct, but the gap between the actual threshold (64W) and the stated assumption (300W) is large. While the authors note they do not optimize the constant, it would be helpful to briefly indicate where the remaining slack is consumed (e.g., in the 100W term of (9) and the 3ℓ bound of Lemma 3.7), so the reader can verify that 300W is sufficient for the final contradiction in the proof of Theorem 1.5.
minor comments (6)
  1. Section 2.1: The set X is defined as {(a,b) : a,b ∈ [4], a ≠ b}, which has 12 elements. The notation [4] is standard but should be stated explicitly (e.g., [4] = {1,2,3,4}) for completeness.
  2. Lemma 4.2: The reachability argument from (1,2) lists reachable vertices in steps. The vertex (2,1) is reached via the walk (1,2)→(2,3)→(3,4)→(4,2)→(2,1), which has length 4, not 3. The text says 'in three steps, we can reach (1,3),(1,4),(3,2),(4,2)' and then separately handles (2,1). This is correct but the organization is slightly confusing; consider grouping by distance or adding a remark that (2,1) requires one additional step.
  3. Figure 1 and Figure 2: The figures are helpful but the labels (a, b, a', b', s, t, z, z', v') could be more clearly matched to the text. In particular, the vertex 's' in Figure 2 is not labeled in the figure but is referenced in the text following Claim 3.10.
  4. Reference [3] (Bucić, Christoph, Pokrovskiy, Steiner) and [4] (Bucić, Hendrey, Mohar, Steiner, Yepremyan) are both cited as arXiv preprints from 2026. The paper should confirm these references are stable (e.g., accepted versions or DOIs) at the time of publication.
  5. Definition 3.4: The term 'w-nearly transitive' is introduced here. It would be helpful to briefly contrast this with the unweighted notion of near-transitivity used in [4], to make clear what the weighted version adds.
  6. Proof of Theorem 1.5: The final computation gives ℓ ≥ √(n/(9·300)) = √n / (30√3). The constant could be simplified to √n/√27000 or stated as Ω(√n) with the explicit constant in a remark.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for a careful reading and for identifying three points where the exposition can be strengthened. All three comments concern clarifications rather than corrections; we are happy to incorporate the suggested additions in the revised manuscript.

read point-by-point responses

  1. Referee: Lemma 3.7: The pairwise internal disjointness of subpaths across non-adjacent indices (which follows from H being induced, t ≥ 4) should be made fully explicit, as it is load-bearing for the claim that Z is a simple cycle.

    Authors: We agree that the pairwise disjointness argument deserves to be stated more explicitly. In the current text, the sentence beginning 'Since H is an induced cycle, every cycle F_i intersects only the two neighboring cycles F_{i-1} and F_{i+1}' is intended to convey this, but the referee is right that the implication for non-adjacent subpaths is not spelled out. We will add two sentences making the argument explicit: because H is induced and t ≥ 4, any two non-adjacent cycles F_i and F_j (with |i−j| ≥ 2, indices taken mod t) are vertex-disjoint in D; consequently, the subpath of F_i from v_i to v_{i+1} can only intersect F_j if j ∈ {i−1, i+1}, and the 'first vertex encountered' choice already handles the adjacent case. This ensures that the subpaths are pairwise internally disjoint, so their concatenation is a simple directed cycle. revision: yes

  2. Referee: Lemma 3.8, Claim 3.9: The referee asks us to verify that the −W in (2) (subtracting w(u')) and the +4W bounds are not double-counting the same vertex weight, and to add a clarifying remark.

    Authors: The referee's observation is correct: the accounting is sound, but a clarifying remark would help the reader. The −W in (2) accounts for the double-counting of the vertex u' when computing w(S'[a', b']) = A + B − w(u'), since u' lies on both subpaths S'[a', u'] and S'[u', b']. The +4W bounds (|A − D_a| ≤ 4W and |B − D_b| ≤ 4W) arise from a different source: they bound the discrepancy between weighted distances along S' and along Q, caused by at most two adjacency steps (a to a', u to u'), each contributing at most 2W. These are indeed distinct quantities—the former is a correction for a shared vertex on a single path, while the latter accounts for the gap between two different geodesic paths. We will add a brief remark to this effect after equation (5). revision: yes

  3. Referee: Lemma 3.8, Claim 3.10: The gap between the actual threshold (64W) and the stated assumption (300W) is large. The referee asks us to briefly indicate where the remaining slack is consumed.

    Authors: We agree that a brief remark tracing the slack would improve readability. The threshold 300W is used at two points in the argument. First, in Claim 3.10, the inequality (11) requires Γ/2 − 32W > 0, i.e., Γ > 64W, to conclude w(T) > w(P). Second, and more importantly, the assumption Γ ≥ 300W is used in the final contradiction of Theorem 1.5 (Section 3.5): it ensures that the weighted-geodesic subpath Q_H guaranteed by Lemma 3.8 has weight w(Q_H) ≥ Γ/2 − 100W > 50ℓ, which strictly exceeds the 3ℓ upper bound from Lemma 3.7. The slack between 64W and 300W is consumed by the 100W term in (9) (arising from the 20W bound of (8) and the initial β = Γ/2 − 80W threshold) and by the need for the lower bound on w(Q_H) to exceed 3ℓ rather than merely being positive. We will add a sentence at the end of the proof of Lemma 3.8, or as a remark following it, summarizing where the constant 300 arises and noting that it is not optimized. revision: yes

Circularity Check

0 steps flagged

No significant circularity found; the paper is self-contained.

full rationale

The paper's two main results, Theorem 1.3 and Theorem 1.5, are derived from scratch without circular dependencies. Theorem 1.3 constructs a specific digraph D (Definition 4.1) and proves its properties (strong connectivity, vertex-transitivity, non-Hamiltonicity via parity, and cycle-length divisibility) through self-contained lemmas (Lemmas 4.2–4.4). The parity obstruction is a new combinatorial argument, not a renaming of a known result. Theorem 1.5 builds on the cycle graph framework of [4] (Bucić et al.), but the key innovation—the weighted cycle graph (Definition 3.3), the weighted shortcut property (Lemma 3.7), and the folding argument (Lemma 3.8)—is new to this paper. Lemma 3.1 is cited from [4] as an external input (the expansion bound ℓ ≥ n/(9d)), used as a black box, not as a self-citation. The authors of the present paper (Li, Methuku) do not overlap with the authors of [4]. The proof of Lemma 3.8 is intricate but self-contained: it defines W, Γ, and the near-transitivity condition, then derives the bound Γ ≤ 300W through a maximality contradiction that does not reduce to its inputs by construction. No step in the derivation chain is equivalent to its inputs by definition, and no self-citation is load-bearing in a circular way.

Axiom & Free-Parameter Ledger

2 free parameters · 3 axioms · 2 invented entities

The paper introduces two main constructs: the weighted cycle graph framework (methodological) and the layered digraph construction (structural). Both are fully defined and their properties are derived from first principles within the paper. The only external dependency is Lemma 3.1 from [4], used as a black-box input. The constant 300 is an artifact of the proof, not a fitted parameter. The layer size 12 is forced by the choice of [4] and the parity requirement.

free parameters (2)
  • Layer size |X| = 12 = 12
    The set X = {(a,b) : a,b ∈ [4], a≠b} has exactly 12 elements. The choice of [4] (hence 12) is fixed by the construction; the parity argument requires an even layer size so that a full cycle is an odd permutation. The constant 1/12 in the perimeter gap is a direct consequence.
  • Constant 300 in Lemma 3.2 = 300
    The constant 300 in the bound ℓ ≥ d/300 arises from the accumulated weight bounds in Lemma 3.8 (constants 100, 80, 20, 10, 7, etc.). The authors explicitly state they make no attempt to optimize it. It is not fitted to data but is a consequence of the proof structure.
axioms (3)
  • domain assumption Every connected vertex-transitive digraph D on n vertices with directed diameter d and circumference ℓ satisfies ℓ ≥ n/(9d) (Lemma 3.1, from Lemmas 2.2 and 2.4 of [4]).
    This is a prior result from Bucić et al. [4] used as input. It is not proved in this paper. It provides the expansion bound that, combined with Lemma 3.2, yields Theorem 1.5.
  • domain assumption The cycle graph C(D) of a connected vertex-transitive digraph D is connected and inherits symmetries from D (following [4]).
    Used implicitly in Section 3.1. The cycle graph construction and its basic properties are from [4]; the paper extends this by adding weights.
  • standard math Every sufficiently large integer is a nonnegative integer combination of 3 and 4.
    Used in Lemma 4.2 to adjust walk lengths modulo m. This is a standard number-theoretic fact (the Frobenius number of {3,4} is 5).
invented entities (2)
  • Weighted cycle graph C(D) with weight w(F) = |F| independent evidence
    purpose: Auxiliary structure for proving the linear diameter-circumference relation (Lemma 3.2). The weighting eliminates the multiplicative loss factor present in the unweighted approach of [4].
    The weighted cycle graph is a methodological tool, not a new mathematical object postulated to exist. Its properties are derived from D. The key property (w-near transitivity, Lemma 3.5) is proved, not assumed. The framework makes a falsifiable prediction: that the weighted diameter is O(ℓ), which is verified in the proof.
  • Cyclic layered digraph D on X × Z_m with X = {(a,b) : a,b ∈ [4], a≠b} independent evidence
    purpose: Construction proving Theorem 1.3 (linear perimeter gap). The layer structure and edge rule encode the parity obstruction.
    The digraph D is explicitly constructed in Definition 4.1. Its properties (vertex-transitivity, strong connectivity, no Hamiltonian cycle, all cycles divisible by m) are all verified in Lemmas 4.2–4.4. No property is assumed without proof.

pith-pipeline@v1.1.0-glm · 19528 in / 3393 out tokens · 366850 ms · 2026-07-08T23:26:54.154993+00:00 · methodology

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read the original abstract

The search for Hamiltonian cycles in vertex-transitive graphs and digraphs is a classical problem at the interface of graph theory and group theory. In the undirected setting, this goes back to famous conjectures of Lov\'asz and Thomassen predicting that every sufficiently large connected vertex-transitive graph is Hamiltonian. The directed analogue has an even richer history, originating with Rankin in 1946, naturally translating the search for long cycles into classical group rearrangement problems. It was shown by Trotter and Erd\H{o}s in 1978 that connected vertex-transitive digraphs need not be Hamiltonian. In 1981, Alspach asked whether there exist connected vertex-transitive digraphs whose longest directed cycle misses arbitrarily many vertices. This question was only recently resolved by Buci\'c, Hendrey, Mohar, Steiner and Yepremyan, who constructed connected vertex-transitive digraphs on $n$ vertices whose longest directed cycle omits $(1-o(1))\log n$ vertices. They further conjectured that the number of omitted vertices can grow linearly with $n$, remarking that it would already be interesting to improve their logarithmic lower bound to a polynomial bound. In this paper, we confirm their conjecture in a strong form by constructing infinitely many connected vertex-transitive digraphs on $n$ vertices whose longest directed cycle omits at least $n/12$ vertices. In the same work, Buci\'c, Hendrey, Mohar, Steiner and Yepremyan also proved that every connected vertex-transitive digraph on $n$ vertices contains a directed cycle of length $\Omega(n^{1/3})$, giving the first lower bound for this problem that grows with $n$. We improve this to $\Omega(\sqrt n)$, matching the order of Babai's classical theorem from 1979 for undirected vertex-transitive graphs.

Figures

Figures reproduced from arXiv: 2607.05807 by Abhishek Methuku, Bowen Li.

Figure 1
Figure 1. Figure 1: Illustration of the paths P, Q, L′ , R′ and S ′ used in the proof of Lemma 3.8. Blue segments highlight vertex pairs whose distance is either zero or one. Claim 3.9. Either every vertex of L ′ ∩ (Q ∪ N(Q)) lies at weighted distance at most 10W from u ′ along S ′ , or every vertex of R′ ∩ (Q ∪ N(Q)) lies at weighted distance at most 10W from u ′ along S ′ [PITH_FULL_IMAGE:figures/full_fig_p008_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: ). Since w(L) ≥ Γ/2−W and ϕ is a weight-preserving automorphism, we have w(L ′ ) ≥ Γ/2−W. Combining this with the upper bound w(L ′ [u ′ , z′ ]) ≤ 10W from (7), and noting that w(L ′′) ≥ w(L ′ ) − w(L ′ [u ′ , z′ ]), we obtain: w(L ′′) ≥ Γ 2 − 11W > β. (10) Claim 3.10. We have L ′′ ∩ (P[x, y] ∪ N(P[x, y])) ̸= ∅. Proof of Claim. Suppose, for contradiction, that L ′′ ∩ (P[x, y] ∪ N(P[x, y])) = ∅. We construc… view at source ↗

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Reference graph

Works this paper leans on

18 extracted references · 18 canonical work pages · 3 internal anchors

  1. [1]

    The search for long paths and cycles in vertex-transitive graphs and digraphs

    B. Alspach. “The search for long paths and cycles in vertex-transitive graphs and digraphs”. In: Combinatorial Mathematics, VIII. Vol. 884. Lecture Notes in Mathematics. Berlin: Springer, 1981, pp. 14–22 (cit. on pp. 1, 2)

    work page 1981

  2. [2]

    Long cycles in vertex-transitive graphs

    L. Babai. “Long cycles in vertex-transitive graphs”. In:Journal of Graph Theory3.3 (1979), pp. 301– 304 (cit. on p. 1)

    work page 1979

  3. [3]

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

    M. Buci´ c, M. Christoph, A. Pokrovskiy, and R. Steiner. “Towards the Lov´ asz Conjecture via Sub- linear Expanders”. In:arXiv preprint arXiv:2606.09742(2026). arXiv:2606.09742(cit. on p. 2)

    work page internal anchor Pith review Pith/arXiv arXiv 2026

  4. [4]

    Long cycles in vertex transitive digraphs

    M. Buci´ c, K. Hendrey, B. Mohar, R. Steiner, and L. Yepremyan. “Long cycles in vertex transitive digraphs”. In:arXiv preprint arXiv:2602.16333(2026) (cit. on pp. 2–8, 11)

    work page arXiv 2026

  5. [5]

    Hamiltonian cycles and paths in Cayley graphs and digraphs—a survey

    S. J. Curran and J. A. Gallian. “Hamiltonian cycles and paths in Cayley graphs and digraphs—a survey”. In:Discrete Mathematics156.1–3 (1996), pp. 1–18 (cit. on p. 1)

    work page 1996

  6. [6]

    Longer cycles in vertex transitive graphs

    M. DeVos.Longer cycles in vertex transitive graphs. Preprint, arXiv:2302.04255. 2023 (cit. on p. 2)

    work page internal anchor Pith review Pith/arXiv arXiv 2023

  7. [7]

    Some theorems on abstract graphs

    G. A. Dirac. “Some theorems on abstract graphs”. In:Proceedings of the London Mathematical Society. 3rd ser. 2 (1952), pp. 69–81 (cit. on p. 1)

    work page 1952

  8. [8]

    Longest cycles in vertex- transitive and highly connected graphs

    C. Groenland, S. Longbrake, R. Steiner, J. Turcotte, and L. Yepremyan. “Longest cycles in vertex- transitive and highly connected graphs”. In:Bulletin of the London Mathematical Society57.10 (2025), pp. 2975–2990 (cit. on p. 2)

    work page 2025

  9. [9]

    Reducibility among combinatorial problems

    R. M. Karp. “Reducibility among combinatorial problems”. In:Complexity of Computer Compu- tations. Ed. by R. E. Miller and J. W. Thatcher. The IBM Research Symposia Series. New York: Plenum Press, 1972, pp. 85–103 (cit. on p. 1)

    work page 1972

  10. [10]

    Problem 11

    L. Lov´ asz. “Problem 11”. In:Combinatorial Structures and Their Applications. New York: Gordon and Breach, 1970, p. 497 (cit. on p. 1)

    work page 1970

  11. [11]

    Intersections of longest cycles in vertex-transitive and highly connected graphs

    J. Ma and Z. Zhao.Intersections of longest cycles in vertex-transitive and highly connected graphs. Preprint, arXiv:2508.17438. 2025 (cit. on p. 2)

    work page internal anchor Pith review Pith/arXiv arXiv 2025

  12. [12]

    Norin, R

    S. Norin, R. Steiner, S. Thomass´ e, and P. Wollan.Small hitting sets for longest paths and cycles. Preprint, to appear in Proceedings of the American Mathematical Society. 2025 (cit. on p. 2)

    work page 2025

  13. [13]

    Hamiltonian paths in Cayley graphs

    I. Pak and R. Radoiˇ ci´ c. “Hamiltonian paths in Cayley graphs”. In:Discrete Mathematics309.17 (2009), pp. 5501–5508 (cit. on p. 1)

    work page 2009

  14. [14]

    A campanological problem in group theory

    R. A. Rankin. “A campanological problem in group theory”. In:Proceedings of the Cambridge Philosophical Society44 (1948), pp. 17–25 (cit. on p. 2)

    work page 1948

  15. [15]

    Cayley color groups and Hamilton lines

    E. Rapaport-Strasser. “Cayley color groups and Hamilton lines”. In:Scripta Mathematica24 (1959), pp. 51–58 (cit. on p. 2)

    work page 1959

  16. [16]

    When the Cartesian product of directed cycles is Hamiltonian

    J. Trotter William T. and P. Erd˝ os. “When the Cartesian product of directed cycles is Hamiltonian”. In:Journal of Graph Theory2.2 (1978), pp. 137–142 (cit. on p. 2)

    work page 1978

  17. [17]

    A. T. White.Graphs, Groups and Surfaces. 2nd ed. Vol. 8. North-Holland Mathematics Studies. Amsterdam: North-Holland, 1984 (cit. on p. 2)

    work page 1984

  18. [18]

    A survey: Hamiltonian cycles in Cayley graphs

    D. Witte and J. A. Gallian. “A survey: Hamiltonian cycles in Cayley graphs”. In:Discrete Mathe- matics51.3 (1984), pp. 293–304 (cit. on p. 1)

    work page 1984