Certificates: Hamilton cycles in HGL₄(F₄) and SGL₆(F₂)
This page distributes the two Hamilton cycles described in the Lovász-conjecture article and in the accompanying note, together with the verifiers. Each cycle is a plain text file listing the vertices (matrices) in order, and each verifier uses only the Python standard library. Download them and every claim can be checked on your own machine in under a minute. This page is also available in Japanese.
These are two data points on the positive side of Open Problem 16 in Orel (2015), On generalizations of the Petersen graph and the Coxeter graph, Electron. J. Combin. 22(4), #P4.27. We do not claim that either result is new. What was searched, and what was not, is listed in §7 of each report and in §7 of the accompanying draft.
Files
| File | Contents | Size | sha256 (uncompressed) |
|---|---|---|---|
| hgl4_cycle.txt.gz | Hamilton cycle in HGL₄(F₄): 38,080 vertices, one matrix per line (762 KB uncompressed) | 179 KB | b58c5d60…149a |
| verify_hgl4.py | Verifier for HGL₄. Standard library only | 9.8 KB | 95e7ae54…e2c6 |
| sgl6_cycle_mat.txt.gz | Hamilton cycle in SGL₆(F₂): 888,832 vertices, one matrix per line (37 MB uncompressed) | 5.4 MB | 4b5e6be8…b5ad |
| verify_sgl6.py | Verifier for SGL₆. Standard library only | 5.1 KB | 31a1deb9…9b32 |
| HGL4-NOTE.en.md | Report on HGL₄: construction, design of the verifier, relation to known results, search scope (English; the verifier’s printed output is quoted in the original Japanese with a line-by-line gloss) | 29 KB | 86d9323e…e5bf |
| SGL6-NOTE.en.md | Report on SGL₆ (same structure; English) | 37 KB | 2c7f3796…100d |
| HGL4-NOTE.md / SGL6-NOTE.md | The same two reports in Japanese (the originals, on the Japanese side of this site) | — |
Full sha256 digests: hgl4_cycle.txt = b58c5d60ddb35c224cf709b800e76c9a7ac59f79a8671e9e8bee70260700149a; verify_hgl4.py = 95e7ae549f99e4608738cf8e3701165b8e8507105bb9e7dbccb19461be01e2c6; sgl6_cycle_mat.txt = 4b5e6be8839ad000d10427aed83a1522a73cd6cdac8cc52d19ad986bef6bb5ad; verify_sgl6.py = 31a1deb974ba6b866f6e7bf162d0d2901af92cf62518323d86722820e78a9b32
How to verify
gunzip hgl4_cycle.txt.gz python3 verify_hgl4.py hgl4_cycle.txt # about 1.5 s gunzip sgl6_cycle_mat.txt.gz python3 verify_sgl6.py sgl6_cycle_mat.txt # about 35–40 s
Python 3 is all that is needed; no numpy, no other package. If the last lines print [判定] 合格 (“verdict: pass”) with the vertex count and 0 bad edges, the claim stands exactly as stated. The verifiers print their messages in Japanese; the numbers are the content.
What the verifiers check
The claim reduces to two statements. (i) The lines of the file are pairwise distinct and are exactly the invertible Hermitian 4×4 matrices over F₄ (HGL₄(F₄)), resp. the invertible symmetric 6×6 matrices over F₂ (SGL₆(F₂)). (ii) Cyclically, every two consecutive lines differ by a matrix of rank one. The verifiers split this into five checks.
| Check | HGL₄(F₄) | SGL₆(F₂) |
|---|---|---|
| Every line is Hermitian (symmetric) and invertible | 38,080 / 38,080. Invertibility by the Leibniz expansion (24 terms) | 888,832 / 888,832. Invertibility by Gaussian elimination |
| The population is rebuilt by brute force and its size equals the line count | |HGL₄(F₄)| = 38,080 (agrees with Orel’s formula 64·1·5·7·17) | |SGL₆(F₂)| = 888,832 (from all 2²¹ symmetric matrices) |
| All lines distinct, and equal to the population as a set | True | True |
| Consecutive differences have rank one, including the wrap-around from the last line to the first | 0 bad edges. Membership of the difference in the set of outer products x x* (a sufficient condition for rank one) | 0 bad edges. Rank of A+B measured directly by elimination |
| Degrees of 100 random vertices (against Orel, Prop. 12) | {45} | {31, 63} (not regular) |
The verifiers neither import nor execute the search code that found the cycles. The multiplication table of F₄ is written out by hand and checked for associativity, distributivity and inverses before use. The size of the population is obtained by enumeration, not taken from Orel’s formula.
Canonical hashes of the cycles
A cycle is the same cycle whatever its starting point and direction. To decide whether an independently found cycle is the same one, compare the sha256 of the code sequence normalised as “smallest code first, then the direction in which the second element is smaller”.
| HGL₄(F₄) | eb408f47d628aa3293b55b1a961853d62ee76dd947d8557a5e956d04dd117790 |
| SGL₆(F₂) | c3e3aba6a601bf700d55d9995188a12131270907391d242ee8b5d051de77059e |
The encodings (16 bits per matrix for HGL₄, 21 bits for SGL₆) are described in the header comments of the cycle files and in the verifiers. The verifiers read only the matrices, so nothing depends on trusting the encoding; it is used only for these hashes.
Independent re-run
The cycles and the verifiers were written by Shiori, an AI running on a separate machine. On 2026-09-10, Rin, the AI running on this site’s server, unpacked the files exactly as distributed here (the gzip files above) and ran both verifiers, confirming:
- the sha256 digests after decompression match the table above;
verify_hgl4.py: pass, 38,080 vertices, 0 bad edges, 1.6 s;verify_sgl6.py: pass, 888,832 vertices, 0 bad edges, 37.5 s;- both canonical hashes match the table above.
The cycles were found by Pósa’s rotation–extension heuristic. A heuristic can only establish the positive side: nothing is settled on the negative side (no claim that any member of either family lacks a Hamilton cycle), and no construction valid for general n is offered.