Markings of a graph that realise kei, symmetric and CNS-quandles
Abstract
A marking of a finite (di)graph Γ with vertex set V is a map R: V → AutΓ; it makes V into a right quasigroup, and by a theorem of Ta that right quasigroup is a rack exactly when R_(Rᵥ(w))=RᵥR_wRᵥ⁻¹ for all v,w. Ta tabulated the numbers μ_(rack)(Γ) and μ_(qnd)(Γ) of markings realising a rack, respectively a quandle, for complete graphs, stars and cycles, and closed his paper by asking for Cayley (di)graphs of racks carrying extra structure, naming symmetric racks. We do not characterise those Cayley graphs; we cross the extra structure named there with the counting problems of the same list, a question we have not found posed anywhere. For a graph Γ we count the markings whose quandle is a kei (μₖₑᵢ), admits a good involution (μ_(sym)), or is a connected noninvolutory symmetric quandle (μ_(CNS)), and we count the pairs consisting of a marking and a good involution of it (μₚₐᵢᵣ). We determine all four counts for Kₙ with n ≤ 7, for the star K_(1,n-1) with n ≤ 7, for the cycle Cₙ with n ≤ 8 and for the vertex-transitive graphs on six vertices; on the way we obtain μ_(qnd)(K₇)=152900, the quandle half of the last undetermined entry of Ta's complete-graph row. Three structural theorems say where the new columns carry no information: μₖₑᵢ=μ_(sym)=μ_(qnd) and μ_(CNS)=0 for every graph all of whose point stabilisers consist of involutions, in particular for every cycle; μ_(CNS)=0 for every graph that is not vertex-transitive, for every star, and for every graph on an odd number of vertices. A fourth says where the μ_(CNS) column lives: if some marking of some graph on n vertices realises a CNS-quandle with good involution ρ, then the same marking realises a CNS-quandle on the perfect matching M_ρ of ρ, on its complement, on Kₙ and on the edgeless graph. At n=6 those four graphs are all of them: exactly 32 of the 32768 labelled graphs on six vertices carry a CNS marking, namely the edgeless graph, the complete graph, the 15 perfect matchings and their 15 complements. So the smallest graph other than a complete or an edgeless one with a nonempty CNS column is 3K₂, and we write a marking of it out. Every theorem and proposition below has been verified with a proof assistant, apart from statements attributed to the literature and two computations marked in the text as external.
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Archived files
- Version 1 · current (opens in a new tab)
Source snapshot 2026-08-30 15:34 UTC
File fingerprint
93c13ea0b06c9c1f6e5bd4d05984df9ebfc32459512fb04e53d62163a8822f33
Claim ledger
Stated results
sgq-01routine2026-08-28
the four counts as cardinalities, and the two decidable forms they need: connectivity (connBᵢff, both directions) and "admits a good involution" (hasGoodInvBᵢff, both directions, via a prefix search with completeness proved)
sgq-02routine2026-08-28
point-stabiliser criterion: if every automorphism of Gamma fixing a vertex is an involution then every quandle marking of Gamma realises a kei, so muₖei = muₛym = mu_qnd and mu_CNS = 0; and every kei marking is symmetric (the identity is a good involution)
sgq-03routine2026-08-28
the cycle satisfies the criterion at every n >= 3 (rigidity: an automorphism of Cₙ fixing two adjacent vertices is the identity), hence muₖei(Cₙ) = muₛym(Cₙ) = mu_qnd(Cₙ) = sigma(n)+1 and mu_CNS(Cₙ) = 0 for every n
sgq-04routine2026-08-28
a marking realising a connected rack forces Gamma vertex-transitive; no marking of a star with n >= 3 realises a connected rack at all, so mu_CNS(K_(1,n-1)) = 0 for every n >= 3
sgq-05candidate2026-08-28
muₖei/muₛym/mu_CNS for Kₙ at n = 0..5: (1,1,0), (1,1,0), (1,1,0), (5,5,0), (26,26,0), (232,272,0)
sgq-06candidate2026-08-28
muₖei/muₛym/mu_CNS for the star K_(1,n-1) at n = 3..7: (2,2,0), (11,11,0), (74,82,0), (962,1072,0), (17572,19306,0)
sgq-07routine2026-08-28
the cycle row of the new columns at n = 3..8, and mu_qnd(C₈) = 16 = sigma(8)+1 one step past the source's table
sgq-08candidate2026-08-28
every CNS-quandle marks a perfect matching: the good involution's own graph Mᵣho is preserved by every row, so one CNS-quandle of order n gives four graphs with mu_CNS > 0 – the edgeless graph, Mᵣho = (n/2)K₂, its complement, and Kₙ
sgq-09candidate2026-08-28
mu(K₆) = (mu_qnd, muₖei, muₛym, mu_CNS) = (6658, 3472, 3922, 30), and the same for the edgeless graph by a theorem rather than a second search
sgq-10candidate2026-08-28
the four columns for the remaining vertex-transitive graphs of order 6: 3K₂ and K_(2,2,2) give (344, 294, 308, 2); 2K₃ and K_(3,3) give (787, 571, 595, 0); C₆ and the prism give (13, 13, 13, 0)
sgq-11candidate2026-08-28
the witness written out: an explicit marking of the perfect matching 3K₂ (and of its complement K_(2,2,2)) realising a CNS-quandle, whose good involution IS that matching
sgq-12routine2026-08-28
negative controls: muₛym(K₅) is neither mu_qnd(K₅) nor muₖei(K₅); mu_CNS(K₆) is not 0, 29 or 31; K₄ fails the point-stabiliser criterion; the star K_(1,2) is not vertex-transitive; C₆ is vertex-transitive with mu_CNS = 0; the tetrahedral quandle is a connected noninvolutory quandle marking of K₄ with no good involution
sgq-13routine2026-08-28
the –load-dynlib route: an import-free mirror of the whole marking search and of the symmetric filters, proved definitionally equal to the originals, plus the bridge allCountsSₑq that carries the four counts
sgq-14routine2026-08-28
muₛymPair defined as Nat.card over pairs (R, rho), its list bridge with Nodup proved, and the five-column evaluator allCounts5
sgq-15candidate2026-08-28
K₇: (mu_qnd, muₖei, muₛym, muₛymPair, mu_CNS) = (152900, 69694, 77632, 523966, 0) – the last? cell of Ta's Table 1, the mu_qnd HALF landed in Lean (Table 1 prints pairs; muᵣack(K₇) = 1164056 is C-only – precision 2026-08-28)
sgq-16candidate2026-08-28
the muₛymPair row for Kₙ (n = 0..6) and for the star K_(1,n-1) (n = 3..7): 1,1,2,11,74,1002,18202 and 6,35,348,5362,131278
sgq-17candidate2026-08-28
the muₛymPair row for the cycle: 11, 30, 57, 152, 373, 1120 at n = 3..8 – the only column of the cycle row that no theorem pins to sigma(n)+1
sgq-18routine2026-08-28
negative controls on the pair column: it is neither muₛym nor mu_qnd, it separates from muₛym already at K₂, and it separates on the cycle row where the other three columns coincide
sgq-19candidate2026-08-28
the CNS column of every graph on n vertices is a filter of the CNS-quandles of order n; at order 6 that decides the column exhaustively – exactly 32 of the 32768 labelled graphs carry a CNS marking, the 1+15+15+1 labellings of K₆-bar, 3K₂, K₂,2,2, K₆
sgq-20routine2026-08-28
mu_CNS(Gamma) = 0 for every graph on an odd number of vertices
sgq-21candidate2026-08-28
the muₛymPair column for the vertex-transitive graphs of order 6: 3K₂/K₂,2,2 1860, 2K₃/K₃,3 3121, C₆/prism 152
Provenance
- Generated by
- Machina Mathematica
- Released by
- Korea Superintelligence Labs
- Source context
- The question. For a (di)graph Γ on n vertices, a marking is a function R: V → Aut Γ; it realises a right quasigroup V_R^Γ, and Lực Ta's Table 1 (arXiv:2506.04437 v5, *J. Non-Assoc. Struct.* 1 (2026), issue 1, jonas:17215) prints (μᵣack(Γ), μ_qnd(Γ)) — how many markings realise a rack, resp. a quandle — for Kₙ, K_(1,n-1) and Cₙ. Ta's *other* paper (arXiv:2505.08090) studies good involutions: an involution ρ of the carrier with ρ sₓ = sₓ ρ and s_(ρ(x)) = sₓ⁻¹, making (Q, ρ) a
- Snapshot
- 2026-09-07 03:53 UTC
- Ledger commit
801848d7