Thurston–Bennequin-symmetrical graphs: the odd-graph conjecture, the Heawood case, and the classification on ten vertices
Abstract
Chau and Shah (arXiv:2603.28487) call a finite simple graph G TB-symmetrical when, for every two cycle lengths r ≠ s, the numbers of r-cycles through each edge, through each pair of adjacent edges and—with a sign recording orientation—through each pair of non-adjacent edges are a common nonnegative multiple ρ_(r,s) of the corresponding numbers for s-cycles. For such G the total Thurston–Bennequin number of every Legendrian embedding is (1+Σ_(r ≠ s)ρ_(r,s)) times the contribution of the s-cycles. They conjecture that every odd graph Oₙ is TB-symmetrical, prove it for n=2,3, and classify the connected pendant-free (almost-)TB-symmetrical graphs on at most nine vertices. We show that the conjecture fails at its first open case: O₄=K(7,3) is not TB-symmetrical, the orientation condition failing between cycle lengths 6 and 7 on two explicit pairs of non-adjacent edges. Two computational statements of the source are corrected: the Heawood graph is TB-symmetrical (the cited code builds a Möbius ladder in its place, and that graph is not even almost-TB-symmetrical), and the 1-clique sum K₅oplus⁰K₅ is missing from the nine-vertex list. An exhaustive computation over the 9 808 209 connected pendant-free graphs on ten vertices, reported outside the formal development, finds 46 almost-TB-symmetrical graphs, 45 of them TB-symmetrical; two are new for the subject: the subdivision S(K₄), a second TB-symmetrical graph with cycles of more than one length that is not obtainable from complete and complete bipartite graphs by the source's operations, and K_(5,5) minus a perfect matching, a second almost- but not TB-symmetrical graph after the cube. Finally the Pappus and Desargues graphs are 3-arc transitive and not TB-symmetrical, so 3-arc transitivity does not imply TB-symmetricity, which closes the natural route to the conjecture. The (almost-)TB-symmetricity statements of the theorems are verified in Lean 4.
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Claim ledger
Stated results
TB0routine2026-09-07
Definition 4.1 of arXiv:2603.28487v1 over Fin n graphs: fE, gP, uCnt as Nat.card of the sets it cuts out, a cycle enumerator with both directions of its membership characterisation and its Nodup proved (mem_walks, mem_cyclesAt, nodup_cyclesAt), and the decision procedure in both directions (tbSymₒf_check, almostTBSymₒf_check, notₜbSymₒfₙap_det, notₐlmostTBSymₒfₑdge_det, notₐlmostTBSymₒfₐrc_det)
TB1candidate2026-09-07
the odd graph O₄ = K(7,3) is NOT TB-symmetrical: wD at the two non-adjacent edge pairs (012,345,013,245) and (012,345,013,246) is (-1, 0) at cycle length 6 and (0, -2) at length 7, a nonzero 2x2 determinant, so clause (3) of Definition 4.1 has no rho for the pair (6,7) in either order
TB2correction2026-09-07
the Heawood graph IS TB-symmetrical, so Proposition 5.3(3) of arXiv:2603.28487v1 is false and the implication diagrams lose their counterexample to '4-arc transitive implies TB-symmetrical'
TB3correction2026-09-07
the Moebius ladder on 14 vertices – a 14-cycle plus its seven antipodal chords, the graph the source's own cited Sage file builds where the Heawood graph was intended – is not even almost-TB-symmetrical: the rim edge lies in one 4-cycle and two 6-cycles, the chord in two of each
TB4candidate2026-09-07
the subdivision S(K₄) (10 vertices, 12 edges, four 6-cycles and three 8-cycles) is TB-symmetrical – a second graph, after the Petersen graph, that is TB-symmetrical and not obtainable from complete graphs, complete bipartite graphs and the operations of Lemmas 6.1-6.3
TB5measurement2026-09-07
the classification on TEN vertices: of the 9,808,209 connected pendant-free graphs on 10 vertices exactly 46 are almost-TB-symmetrical and 45 of those are TB-symmetrical; 35 of the 46 have all cycles of one size, and the other eleven are K₁0, K_(3,7), K_(4,6), K_(5,5), K₄(+)³K₄, K₅(+)¹K₅, the two triple 1-clique sums of K₄, S(K₄), the Petersen graph, and K_(5,5) minus a perfect matching
This ledger entry is reported in prose and is not bound to a Lean theorem.TB6candidate2026-09-07
K_(5,5) minus a perfect matching (the 5-crown) is almost-TB-symmetrical but not TB-symmetrical – the second such graph after Q₃, and the only one on ten vertices
TB7routine2026-09-07
the compute-first gate: Proposition 5.3(1),(2),(4),(5) of arXiv:2603.28487v1 and Theorem 4.6 at K₄, K₅, K_(3,3) reproduced from Definition 4.1
TB8routine2026-09-07
negative controls: a too-large claim refuted (not every graph is (almost-)TB-symmetrical; almost does not imply full; the decision procedure is not constantly true), a too-small one refuted (TB-symmetricity is not the trivial case of Lemma 4.2, and does not force regularity or vertex-transitivity), non-vacuity (the index lists are nonempty, the counts are nonzero, and clause (3) is NOT identically zero on the graphs proved TB-symmetrical), and every computed count pinned against a neighbouring value
TB9correction2026-09-07
K₅ (+)⁰ K₅, the 1-clique sum of two copies of K₅ (nine vertices, twenty edges), is TB-symmetrical and is missing from the list of Theorem 6.4 of arXiv:2603.28487v1
TB10measurement2026-09-07
subdividing every edge appears to preserve (almost-)TB-symmetricity in both directions: nine of nine tested graphs agree, including the two negatives K_(2,2,2) (neither) and Q₃, K_(5,5)-PM (almost only)
This ledger entry is reported in prose and is not bound to a Lean theorem.TB11candidate2026-09-07
3-arc transitivity does not imply TB-symmetricity: the Pappus graph and the Desargues graph are 3-arc-regular and NOT TB-symmetrical, while the Petersen graph (3-arc-regular) and the Heawood graph (4-arc-regular) are
Provenance
- Generated by
- Machina Mathematica
- Released by
- Korea Superintelligence Labs
- Source context
- A Legendrian graph in the standard contact 3-space has a total Thurston–Bennequin invariant TB(G̃) = Σ_γ tb(γ), summed over the cycles of G. Chau and Shah (arXiv:2603.28487v1, 30 Mar 2026) isolate the combinatorial condition on the *abstract* graph under which that sum collapses to the s-cycles alone, and this family is about that condition and nothing else — no contact geometry is formalised here.
- Snapshot
- 2026-09-07 03:53 UTC
- Ledger commit
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