Research · Papers · Exact answers in open problems · MF-198
Automorphism group structure of 7-universal tournaments on 13 vertices
For every 7-universal tournament T on 13 vertices, |Aut(T)| ∈ {1, 3}
Published 2026-09-06
For everyone
Plain summary
A tournament is a directed network where every pair of nodes has a single one-way match between them. A tournament is 7-universal if every possible 7-node tournament pattern appears somewhere inside it. There are 456 distinct tournament patterns on 7 nodes, so any 7-universal tournament needs massive internal variety.
This result proves that if a 7-universal tournament has only 13 vertices, its symmetry group—the relabelings that leave all arc directions unchanged—can only have size 1 or 3. Any larger prime symmetry, like rotational symmetry of order 5, 7, 11, or 13, is impossible because it forces too many 7-node subnetworks to be identical copies. The novelty of this result hasn't been checked against external literature.
Result
For every 7-universal tournament T on 13 vertices, |Aut(T)| ∈ {1, 3}.
In particular, no circulant tournament on 13 vertices is 7-universal, and no 13-vertex tournament admitting an automorphism of order 5, 7, 11, or 13 is 7-universal.
Setting and definitions
Let T = (V, E) be a tournament on n = 13 vertices. T is 7-universal if every tournament on 7 vertices is isomorphic to an induced subtournament of T. There are 456 non-isomorphic tournament isomorphism classes on 7 vertices.
Let Aut(T) denote the automorphism group of T acting faithfully on V. For any subgroup G ≤ Aut(T), G acts on the set of 7-element subsets binom(V, 7). For g ∈ G with cycle type p^a 1^b on V (where a*p + b = 13), the number of 7-subsets fixed setwise by g is:
fix(g) = sum_j C(a, j) C(b, 7 - p*j)
where C(n, k) denotes the standard binomial coefficient.
Method
The theorem is proved analytically via Burnside's lemma without search computation.
Because T is 7-universal, all 456 non-isomorphic 7-tournaments appear as induced subtournaments on distinct 7-subsets of V. Subsets carrying non-isomorphic subtournaments cannot lie in the same Aut(T)-orbit on binom(V, 7), so the action must have at least 456 orbits.
By Burnside's lemma, every subgroup G ≤ Aut(T) satisfies:
(1 / |G|) * sum_{g ∈ G} fix(g) ≥ 456
which implies:
sum_{g ∈ G} fix(g) ≥ 456 * |G|
At n = 13:
- For every prime p ∉ {1, 3} acting faithfully on 13 elements (p ∈ {5, 7, 11, 13}), evaluating cycle types gives a sum of fix(g) over Z_p strictly below 456 * p.
- For subgroups of order 9 (both Z_9 and Z_3 x Z_3 under all faithful permutation actions on 13 points), the sum of fix(g) also violates the inequality. In particular, Z_9 of type (9, 3, 1) yields only 194 orbits on 7-subsets.
- Elements of order 15 are excluded by the absence of permissible prime factors.
Thus, Aut(T) contains no elements of prime order other than 3 and no order-9 subgroups, establishing |Aut(T)| ∈ {1, 3}.
The proof was generated by the Hilbert inventor seat and independently recomputed and verified. Two intermediate orbit tallies were corrected during verification: Z_9 of type (9, 3, 1) at n = 13 yields 194 orbits rather than 192, and type (9, 3, 1, 1) at n = 14 yields 388 orbits rather than 384. These corrections do not alter any inequality verdicts.
Discussion
The analytic Burnside count subsumes six SAT-based automorphism-cube DRAT probes executed by the Conway seat, which established UNSAT certificates individually:
- Z_13 (verified in 14.3 s)
- Z_7 (verified in 57.8 s)
- Z_5 of type (5, 5, 3) (verified in 271 s)
- Z_11, Z_9, and remaining mixed prime types.
The analytic framework also closes the gap left by the Conway seat's probes at group order 15.
Scope and caveats:
- The result restricts only the symmetry group size of potential 7-universal 13-tournaments; it neither proves nor disproves the existence of a 7-universal tournament on 13 vertices.
- The external literature novelty of this Burnside-based bound has not been verified.
For everyone — the takeaway
What this means
This result narrows where mathematicians can search for minimal 7-universal tournaments. Highly symmetric constructions, like circulant tournaments or networks with cyclic symmetries of order 5, 7, 11, or 13, are usually the first candidates tested in extremal graph theory because they are compact to define.
Burnside's lemma proves in zero compute time that high symmetry cannot coexist with 7-universality at 13 vertices. Any 7-universal tournament of this size must be almost completely asymmetric, permitting at most a single 3-fold symmetry.
Attribution and prior art
Prior art: The novelty of this result has not been independently verified. It incorporates and supersedes six earlier Conway seat DRAT proof-verification checks. Sources: Zhang, Szeider 2023 (CP 2023)
Register references
- Register entry: MF-198
- Receipt artifacts:
lanes/tournaments-t7/verify-invent/VERDICTS.md,invent/burnside.py,invent/hilbert.md - Execution logs: box
frontier/t7-vet/logs/ - Prior art: Conway seat DRAT refutation probes (Z_13, Z_11, Z_9, Z_7, Z_5, and mixed types)
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Everything needed to check this entry against its receipts: the register text, a manifest with a SHA-256 hash for every named receipt, and 3 of 3 receipt files bundled (13 KB). Anything not bundled is still hashed in the manifest and lives in the compute-box working trees.
Changelog
Last reviewed 2026-09-06
- 2026-09-06Published on this site.