# HGL₄(F₄) has a Hamilton cycle — the case n = 4 of Orel's Open Problem 16 (the positive side)

**Written by**: a working record[^1] (2026-09-07)
**Work it is based on**: a working record (a working session[^2] — finding the cycle), a working record (a working session — independent verification and the certificate)
**Status**: **this is a document submitted for examination, not a declaration that a record has been updated. It does not claim "world first".**
This document performs no new search. All figures are copied from a single run of `verify_hgl4.py`
(`verify_hgl4.out`, 2026-09-07, 1.4 seconds).
**In the ledger[^3] it has been assigned a record number[^4] after examination**, but the record number is the
ledger-internal fact that "the cycle has become an independently verified certificate"; it is not a claim
of priority in the literature (§7.2).
**Whether to report this outside, and if so how it should be worded, is the operator's decision.**

> **English summary is in §8.**

---

## 1. The claim

### 1.1 The claim (in one line)

> **The graph HGL₄(F₄) (38,080 vertices, 45-regular) has a Hamilton cycle.**
> The evidence is the 38,080 lines of `hgl4_cycle.txt`,
> and the check finishes in **1.4 seconds** with a single Python program that uses the standard library only (`verify_hgl4.py`).

### 1.2 What is not claimed (stated first)

- **This is not "better than Orel".** Orel found Hamilton cycles in the three cases on 280, 448 and 13,888 vertices
  with the Concorde TSP solver, and stated the general case as an open problem. What is used here is Pósa's rotation plus extension, a **heuristic**,
  and a heuristic can only be used on the positive side. The same tool cannot say "there is none".
- **No construction for general n is given.** The second half of Open Problem 16
  ("is there a general way, valid for each n, to construct it?") is not addressed.
- **Nothing is added on the SGLₙ(F₂) side.** The smallest open case there is SGL₆(F₂) (888,832 vertices),
  which in the measurements of a working record does not close (§7).
- **Vertex-transitivity is taken from Orel Prop. 12.** What this verifier confirmed independently
  goes only as far as "the degree of 100 randomly chosen vertices is constantly 45".
- **We do not write "world first".** What can be written is only "**within the search range of this
  ledger, no existing source was found**". The range searched, the queries, and the **searches not
  performed** are listed in §7.2.
  **Whether it may be reported is the operator's decision, and this document does not decide it.**

---

## 2. Definitions (verbatim from the printed paper)

Orel, *On generalizations of the Petersen graph and the Coxeter graph*,
**Electronic Journal of Combinatorics 22(4) (2015), #P4.27**
(Submitted: Sep 27, 2013; Accepted: Nov 4, 2015; Published: Nov 13, 2015.
MSC 2010: 05C50, 15B33, 15B57).

### 2.1 Definition of the graphs (verbatim, corresponding to p.4 of the printed paper)

> Let
> H_n(F₄), HGL_n(F₄), S_n(F₂), SGL_n(F₂)　　(2)
> denote, in the same order, the sets of all hermitian n × n matrices over F₄, invertible
> hermitian n × n matrices over F₄, symmetric n × n matrices over F₂, and invertible
> symmetric n × n matrices over F₂. Each of these four sets is a vertex set of a graph, where
> {A, B} is an edge if and only if rk(A − B) = 1.　　(3)
> We slightly abuse the notation and denote these graphs with the same symbols (2).

And then (in the same paragraph):

> Graphs HGL₂(F₄) and SGL₃(F₂) are the Petersen and the Coxeter graph, respectively,
> as it can be seen in Figures 1 and 2.

**⟹ HGL₄(F₄) is the third step of the Petersen graph** (n = 2, 3, 4 give 10, 280, 38,080 vertices).

### 2.2 Open Problem 16 (verbatim, corresponding to p.14 of the printed paper)

> There are only five known connected vertex-transitive graphs without a Hamilton
> cycle: the complete graph on two vertices, the Petersen graph, the Coxeter graph, and
> two graphs derived from the Petersen and Coxeter graphs by replacing each vertex with
> a triangle (see the survey paper [9] and the references therein). It is believed by many
> mathematicians that no other exist. Since the graphs HGL_n(F₄) and SGL_{2n−1}(F₂) are
> vertex-transitive by Proposition 12 and they generalize the Petersen and the Coxeter
> graph in a natural way, it seems worthwhile to check if these graphs have a Hamilton cycle.
> Unfortunately we did not succeed in this task. With an application of the Concorde TSP
> Solver [5] we were able to find a Hamilton cycle in HGL₃(F₄), SGL₄(F₂), and SGL₅(F₂),
> which are graphs on 280, 448, and 13888 vertices, respectively. We state the general case,
> which include non-vertex-transitive graphs SGL_{2n}(F₂), as an open problem.
>
> **Open Problem 16.** Do graphs HGL_n(F₄) and SGL_n(F₂) contain a Hamilton cycle? If
> the answer is positive, is there a general way, valid for each n, to construct it?

**⟹ Orel's families are just the two, HGLₙ(F₄) and SGLₙ(F₂)** (self-correction N61-1[^5] of a working session).
HGL needs the involution x ↦ x² of F₄/F₂ for the definition of "hermitian", so **there is no HGL over F₂**.

### 2.3 Cardinality and degree (the formulas of the printed paper)

Cardinality (corresponding to p.11 of the printed paper; by [20, Corollary 5.32] and [2, Theorem 9.5.7]):

> |HGL_n(F₄)| = Π_{j=0}^{n−1} (2^{2n} − 2^{2j}) / (2^{j} (2^{j+1} + (−1)^j))
>             = 2^{n(n−1)/2} Π_{j=1}^{n} (2^j + (−1)^j)

- n=2 → 2 · 1 · 5 = **10** (Petersen)
- n=3 → 8 · 1 · 5 · 7 = **280**
- **n=4 → 64 · 1 · 5 · 7 · 17 = 38,080**

Degree (**Proposition 12**, verbatim):

> **Proposition 12.** Let n ≥ 2. Graph HGL_n(F₄) is arc-transitive of degree
> (2ⁿ − (−1)ⁿ)(2^{n−1} − (−1)^{n−1}) / 3.　　(23)
> If n is odd, then SGL_n(F₂) is arc-transitive. If n is even, then SGL_n(F₂) is not vertex-transitive.

- n=4 → 15 · 9 / 3 = **45**

**The exhaustive computation of the verifier reproduces these two independently** ([2] and [6] of §5).

(The OCR of the printed paper is the OCR text in the working directory; the PDF is `orel_ejc22.pdf`
in the same place. The verbatim passages above were copied from the OCR, with the mathematical symbols
(⋆, †, ≥, −) restored to what the printed page means.
Only the glyphs of the symbols were rewritten; the word order and the wording were not changed.)

---

## 3. How the cycle was constructed (how it was found)

**A working record (a working session).** Neither Concorde nor a SAT solver is used.

1. **Encoding**: the set H₄(F₄) of all hermitian matrices is an additive group. The 4 diagonal entries
   lie in F₂ (1 bit each), the 6 off-diagonal pairs in F₄ (2 bits each), 16 bits in total.
   `|H₄(F₄)| = 2⁴ · 4⁶ = 65,536 = 2¹⁶`. **Addition is exactly the bitwise XOR of the codes.**
2. **Neighbourhood**: adjacency is determined by the rank of the difference alone, so once the set `S`
   of rank-1 hermitian matrices (|S| = 85) has been built, the neighbours of a vertex `v` are obtained
   simply by filtering `v XOR S` for invertibility (no adjacency list is held).
   That is, **HGL₄(F₄) is the subgraph of a Cayley graph on H₄(F₄) induced on the invertible vertices**.
3. **Search**: Pósa's rotation plus extension. When the path can no longer be extended from an endpoint,
   a neighbour of that endpoint is chosen and the path is reversed. Closing into a cycle is checked by
   looking at the adjacency of the two ends after the path covers all vertices.
   With 38,080 vertices of valency 45, a cycle was reached in **0.8 seconds**.

**On the reproducibility of this search**: Pósa involves randomisation, so the same cycle will not
necessarily appear again. **What has to be reproduced is not the cycle itself but the fact that a cycle
exists**, and that is what `hgl4_cycle.txt` and `verify_hgl4.py` are delivered for. So that one can
decide whether a cycle found by someone else is the same one,
**a normalised hash independent of rotation and reflection** is given in §6.

---

## 4. Deliverables

| File | Contents | Size |
|---|---|---|
| **`hgl4_cycle.txt`** | The cycle itself. **One line = one vertex = the 4×4 hermitian matrix written row-major as 16 elements of F₄** (separated by spaces into groups of four, one group per matrix row). F₄ = {0,1,2,3} = {0, 1, x, x+1}, x² = x+1 | 38,085 lines (5 of them comments) / 762,110 bytes |
| **`verify_hgl4.py`** | The independent verifier. **Depends only on the Python 3 standard library** (no numpy). It reads `hgl4_cycle.txt` and nothing else | 252 lines |
| `verify_hgl4.out` | The output of the run above (2026-09-07, 1.4 seconds) | — |
| `hgl4_cycle.npy` | The original sequence of 16-bit codes (saved by a working record; for reference) | 152,448 bytes |
| `n924_dump.py` | The `.npy` → `.txt` conversion (a tool for producing the deliverables, not a verifier) | — |

**Usage**:

```bash
python3 verify_hgl4.py hgl4_cycle.txt      # no argument = same-named file
# exit code 0 = pass, 1 = fail
```

---

## 5. The verification procedure (what `verify_hgl4.py` does)

Only **two things** are checked. These two exhaust the claim.

> **(i)** The 38,080 lines of the file are distinct and **exactly exhaust** HGL₄(F₄),
> the set of all invertible hermitian 4×4 matrices.
> **(ii)** The differences of cyclically adjacent lines **all have rank 1**.

### 5.1 The decision procedure is on a different path from the other implementations

The implementations of the earlier working sessions (`lovasz896.py` = elimination, `lovasz896v.py` = 2×2
minors, `r909_verify.py` = the numpy version) are **neither imported nor exec'd**. There are four differences:

| | The two of a working record | `verify_hgl4.py` |
|---|---|---|
| Multiplication in F₄ | carry-less multiplication / a polynomial expression | **a hand-written 4×4 multiplication table** (checked exhaustively for associativity, distributivity, commutativity and inverses before use) |
| Invertibility | elimination / 2×2 minors | **Leibniz expansion** (the 24 permutations of S₄; no signs, since the characteristic is 2) |
| Adjacency | measure the rank | **no rank is measured** — decided by whether the difference D belongs to the set S of outer products |
| Computing the difference | XOR of the codes | **decode to matrices, then subtract entrywise** |
| Input | `.npy` (depends on numpy) | **`.txt` (the matrices themselves)** |

**The logic of the adjacency test (this is the crux)**:
put `S := { x x* : x ∈ F₄⁴, x ≠ 0 }` (the (i,j) entry of `x x*` is `x_i · conj(x_j)`).
If `D = x x*` with x ≠ 0, then every column of D is a multiple of x, so `rk(D) = 1`.
That is, **`D ∈ S` is a sufficient condition for `rk(D) = 1`**, and this is enough to prove an edge.
The converse ("every rank-1 hermitian matrix is of the form `x x*`") is **not used**.
That each element of S is nonzero and has all 2×2 minors 0 is checked exhaustively.

### 5.2 Result of the run (verbatim from `verify_hgl4.out`)

```
[0] 入力 hgl4_cycle.txt : 38080 行
[1] エルミートである行 = 38080 / 38080
[1] 可逆である行（ライプニッツ展開） = 38080 / 38080
[2] |H_4(F_4)| = 65536   (= 2^4 · 4^6)
[2] |HGL_4(F_4)| = 38080   (Orel 2015 の公式 ... は n=4 で 64·1·5·7·17 = 38080)
[3] 行数 = |HGL_4(F_4)| : True
[3] 全行が相異なる      : True
[3] 行の集合 = HGL_4(F_4) : True
[4] 外積 x x^* の相異なる値 |S| = 85   (= (4^4-1)/(4-1) = 85)
[4] S の元で「非零かつ全 2x2 小行列式 0」を破るもの = 0 個  ⇒ S の元の階数はちょうど 1
[5] 閉路の辺の総数（末尾→先頭を含む） = 38080
[5] 差 D = A − B が外積 x x^* の形でない辺 = 0 本
[5] 使われた階数 1 の元の種類 = 85 / 85
[6] 無作為 100 頂点の次数の集合 = [45]   (Orel Prop.12: 15·9/3 = 45)

[判定] **合格**  頂点 38080 / 行 38080 / 相異なる True / 集合一致 True / 悪い辺 0
所要 1.4 秒
```

The verifier prints its report in Japanese; the block above is left exactly as the program emits it.
Read line by line: `[0]` names the input file and its line count (行 = lines);
`[1]` gives how many lines are hermitian and how many are invertible (by Leibniz expansion);
`[2]` gives the sizes of H₄(F₄) and of HGL₄(F₄), the latter with a reminder of the formula of Orel 2015;
`[3]` states whether the number of lines equals |HGL₄(F₄)|, whether all lines are distinct, and whether
the set of lines equals HGL₄(F₄); `[4]` gives the number of distinct outer products x x* and the number
of elements of S that violate "nonzero and all 2×2 minors zero", from which the elements of S have rank
exactly one; `[5]` gives the total number of cycle edges (including the wrap-around from the last line
to the first), the number of edges whose difference D = A − B is not of the outer-product form, and how
many kinds of rank-one element were used; `[6]` gives the set of degrees of randomly sampled vertices.
The verdict line reads 合格 = **pass**, and 所要 = elapsed time.

**Cross-check against the printed paper**: `|HGL₄(F₄)| = 38,080` and the degree 45 both agree with the
formulas of §2.3. The verifier does not take these two from the printed paper but
**produces them independently by exhaustive computation**.

---

## 6. Where the code is, and its hashes

All of it is in the working directory.

| Item | sha256 |
|---|---|
| `verify_hgl4.py` | `95e7ae549f99e4608738cf8e3701165b8e8507105bb9e7dbccb19461be01e2c6` |
| `hgl4_cycle.txt` | `b58c5d60ddb35c224cf709b800e76c9a7ac59f79a8671e9e8bee70260700149a` |
| `n924_dump.py` | `d51f81e7ba12a53e6e8bab5e236c5253ee1a561f35f0e5aa8688a032d5e46944` |
| `hgl4_cycle.npy` (for reference) | `34ddeeb27bd2aad54d082b69a7a8779713eaf3f8e1b986e59dfded868cc34d57` |

**The three hashes of the cycle itself** (printed by `verify_hgl4.py`):

| Item | sha256 |
|---|---|
| the byte sequence of `hgl4_cycle.txt` | `b58c5d60ddb35c224cf709b800e76c9a7ac59f79a8671e9e8bee70260700149a` |
| the 16-bit codes as comma-separated decimals (in storage order) | `3c00b73d19892e60c2634fbc4c205c4e1e93580c6273eaac3074cee850140096` |
| **the code sequence normalised for rotation and reflection** | **`eb408f47d628aa3293b55b1a961853d62ee76dd947d8557a5e956d04dd117790`** |

The normalisation is "place the smallest code first, and take the direction in which the second element
is smaller". **If the cycle itself is the same, this value agrees regardless of the direction of storage
and of the starting point.** When a third party finds another cycle, this one line decides whether it is
the same one.

**Proofreading**: the latter two agree **digit for digit** with the values produced by the independent
verifier of a working record (the numpy version, which reads the `.npy`). That is, the `.npy` → `.txt`
conversion (`n924_dump.py`) loses no information.

**The convention for the 16-bit code** (a convention for turning the `.txt` back into codes; it is not
part of the verification):

- bit `i` (i = 0..3): the diagonal entry `A[i][i] ∈ F₂`
- bits `4+2k`, `4+2k+1` (k = 0..5, `PAIRS = [(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)]`):
  `A[PAIRS[k]] ∈ F₄` (low bit first)
- `A[j][i]` is determined as `conj(A[i][j])`

---

## 7. Relation to what is known

### 7.1 Orel's table, and the one line this document adds

| Graph | Vertices | Vertex-transitive | Hamilton cycle | Source |
|---|---|---|---|---|
| HGL₂(F₄) = Petersen | 10 | yes | **none** (classical) | — |
| SGL₃(F₂) = Coxeter | 28 | yes | **none** (classical) | — |
| HGL₃(F₄) | 280 | yes | **exists** | Orel 2015 (Concorde) |
| SGL₄(F₂) | 448 | no (n even) | **exists** | Orel 2015 (Concorde) |
| SGL₅(F₂) | 13,888 | yes | **exists** | Orel 2015 (Concorde) |
| **HGL₄(F₄)** | **38,080** | yes | **exists** | **this document (Pósa, a working record / verification by a working record, a working record)** |
| SGL₆(F₂) | 888,832 | no (n even) | open (at the time of this report) | up to 99.9637% in a working record (§7.3). **Closed afterwards in a separate report (SGL6-NOTE)** |
| HGL₅(F₄) | 18,887,680 | yes | open | — |
| SGL₇(F₂) | 112,881,664 | yes | open | — |

**⟹ What this document adds is only the single boldface line of the table above.**

### 7.2 Prior art — **the range searched is stated first**

**This document does not write "world first".** All that can be written is the following one sentence.

> **Within the search range of this ledger, no printed source giving the Hamiltonicity of HGL₄(F₄) was found.**

That "search range" is made explicit, in a form that lets the examination retrace the same steps.

**The search actually performed (2026-09-07)**: a general web search with the following five queries.

1. `"Orel Open Problem 16"`
2. `"HGL_n(F_4) Hamilton cycle"`
3. `"Grassmann graph Hamiltonian"`
4. `"invertible hermitian matrices graph Hamiltonian"`
5. `"Petersen graph" "Coxeter graph" generalizations "Hamilton cycle"`

**Sources found**: only Orel 2015 itself and two papers by the same author on adjacency-preserving maps
(arXiv:1307.3482 / 1307.3484).
**No source stating that Open Problem 16 has been solved came up.**

**Searches performed (a working record, 2026-09-08; transcribed by the examination of a working session)**:

| Means | Result |
|---|---|
| **Going through the complete citation list of Orel 2015** | **Done.** On OpenAlex (`W2172989702`, `cited_by_count` = 6), **all 6** were read down to title and abstract, and **none of them touches Hamiltonicity**. The breakdown is 4 on cores / complement prisms, 1 on rank-1 non-increasing maps, 1 on graph homomorphisms. Among them, **Orel–Višnjić, *Homomorphisms from the Coxeter graph*, Linear Algebra Appl. 2025** treats the same family `Γ_n = SGL_n(F₂)` in 2025 and yet **does not touch Open Problem 16**. **Semantic Scholar could not be retrieved (HTTP 429)** (unauthenticated, 3 retries), and **Crossref returns only a count (`is-referenced-by-count` = 3)**, not a list. The raw responses are in the citation folder in the working directory, the readout in the search output. **The examination recounted the stored JSON with a separate implementation and reproduced 6 items, 0 mentions of Hamilton, DOI 10.37236/3759 and a citation count of 3** (the check script / `.out`) |

**⟹ This one line raises the quality of "was not found" by one step** — a member of the family published
a paper on the same family in 2025, and no settlement of Hamiltonicity is written there.
**Even so, we do not write "world first".**
**Note also that the citation count being as low as 6 is not by itself evidence of difficulty**
(the reverse side of Wall 19[^6]: obscurity and difficulty are different things).
**What was obtained goes only as far as the abstracts, not the bodies.**
The possibility that a single line in the body touches on it has not been eliminated.

**Searches not performed (= the holes in this claim; filling them is the work of the examination)**:

| Means | Status |
|---|---|
| **MathSciNet** (Mathematical Reviews) | **There is no means of access in this environment** (a subscription is required) |
| **Zentralblatt MATH** | Same as above |
| **Full-text search of arXiv** (running `math.CO` through "HGL" / "hermitian" × "Hamilton") | **Not done.** Queries 1–5 above are a general web search and do not consult the arXiv full-text index directly (**a working record ran 7 queries through the arXiv API index — titles, abstracts and authors — and the only hit for `SGL_n` / `HGL_n` was arXiv:2404.17067, with no occurrence of the word Hamilton. That is not a full-text index, so this line stays as it is**) |
| **Scanning the tables of contents of related issues and proceedings after EJC** | **Not done** |
| Non-English literature | **Not done** |

⟹ **Therefore we write not "there is none" but "was not found"** (the reverse side of Wall 19).
**The remaining holes are: MathSciNet, zbMATH, the arXiv full-text search, the scanning of tables
of contents of proceedings and the like, and non-English literature.**

**Whether it may be reported is the operator's decision.** Whether this document goes outside,
and if so in what range, is decided neither by this working session nor by this ledger.

(Its position in the ledger: it has been assigned a **record number** by the examination. The record
number is the ledger-internal fact that "the cycle in HGL₄(F₄) has become an independently verified
certificate"; it is not a claim of priority in the literature.)

### 7.3 The next open case

**SGL₆(F₂) (888,832 vertices)** is the smallest open case on the **non-vertex-transitive** side of
Orel's families (for even n, SGLₙ(F₂) is not even regular — degree 31 for non-alternate and degree 63
for alternate matrices).
The measurement of a working record (a separate machine, Pósa rotation plus extension,
**998 seconds of search**) gives
**a coverage of 99.9637% (888,509 / 888,832; 323 vertices left out), and it does not close**.
(After this report, a search with a changed design did close it; the cycle and its verification are in the companion report SGL6-NOTE.)
The 323 vertices left out are **all non-alternate (degree 31)**; not a single alternate matrix
(degree 63, 1.5625% of the whole) remains, and **317 of the 323 are pairwise non-adjacent**.
**Cost is a property of the procedure** (a working record, wall candidate K59-W1[^7]), so we write not
"SGL₆(F₂) is hard" but "with this Pósa implementation (the representation that reverses the path)
it does not fit in this budget".

---

## 8. English summary

**Claim.** The graph HGL₄(F₄) — vertices: the 38,080 invertible Hermitian 4×4 matrices
over F₄; edges: {A,B} with rk(A−B)=1 — **has a Hamilton cycle.**
This is the case n=4 of the first family in Orel's Open Problem 16
(*Electron. J. Combin.* **22**(4) (2015), #P4.27), which Orel left open; with the
Concorde TSP Solver he found Hamilton cycles in the smaller cases HGL₃(F₄), SGL₄(F₂) and SGL₅(F₂).

**Certificate.** `hgl4_cycle.txt`: 38,080 lines, one per vertex, each line the 4×4
Hermitian matrix written row-major as 16 elements of F₄ = {0,1,2,3} = {0, 1, x, x+1},
x² = x+1, in four space-separated groups of four.

**Verifier.** `verify_hgl4.py` — **Python 3 standard library only**, reads only that text
file, runs in **1.4 seconds**. It checks exactly two things:
(i) the 38,080 lines are distinct and exhaust the set of invertible Hermitian matrices;
(ii) all 38,080 cyclic differences have rank 1.
Independence: F₄ multiplication from a hand-written table (verified against the field
axioms), invertibility by **Leibniz expansion** over S₄ (no elimination, no minors),
adjacency by membership of the difference in S = {x x* : x ≠ 0} (**no rank computation**;
D ∈ S ⟹ rk D = 1 is used, the converse is not), differences by componentwise subtraction
of decoded matrices.

**Canonical hash** (rotation- and reflection-invariant, so a third party who finds another
cycle can tell whether it is the same one):
`eb408f47d628aa3293b55b1a961853d62ee76dd947d8557a5e956d04dd117790`.

**What is not claimed.** This is a heuristic search (Pósa rotation–extension), so it can
only settle the positive side; it is not "better than Concorde". No general construction
for arbitrary n is given. Nothing is added on the SGL_n(F₂) side.

**Priority is explicitly not claimed.** We do **not** say this is new. What we can say is:
*within the search we actually performed, we found no printed source giving a Hamilton
cycle in HGL₄(F₄).* That search was a general web search on 2026-09-07 with five queries
("Orel Open Problem 16"; "HGL_n(F_4) Hamilton cycle"; "Grassmann graph Hamiltonian";
"invertible hermitian matrices graph Hamiltonian"; "Petersen graph" "Coxeter graph"
generalizations "Hamilton cycle"), which returned only Orel 2015 itself and two papers by
the same author (arXiv:1307.3482, arXiv:1307.3484). We **did** go through the complete
citation list of Orel 2015 (6 works on OpenAlex, read down to title and abstract; none of
them touches Hamiltonicity). We did **not** query MathSciNet or zbMATH (no access), did
**not** run an arXiv full-text search, did **not** scan the tables of contents of related
issues and proceedings, and did **not** search non-English literature (§7.2).
Whether and how this note is circulated is the operator's decision, not ours.

---

## 9. Assumptions that remain

1. **Vertex-transitivity and arc-transitivity are taken from Orel Prop. 12.**
   What this verifier confirmed independently goes only as far as "the degree of 100 randomly chosen
   vertices is constantly 45".
   However, **the claim (the existence of a Hamilton cycle) does not depend on this** —
   (i) and (ii) of §5 alone imply that it is a cycle.
2. **The `.npy` → `.txt` conversion was written by this working session.** Proofreading is secured by
   the agreement of the two hashes, but the converter and the verifier have the same author
   (a working session). **A shared conceptual misunderstanding cannot be ruled out.**
   — However, the encoding convention is printed in §6, so a third party can re-read the contents of
   the verifier from the `.txt` alone.
3. **The verbatim passages from the printed paper were copied from the OCR** (the OCR text in the working directory).
   The mathematical symbols were restored to what the printed page means. The PDF (`orel_ejc22.pdf`)
   is in the same place, so at examination it should be checked against the corresponding pages of the PDF.
4. **It has not been through a third-party examination.** The operator, the other lines and outside
   parties have all yet to do so.
5. **The basis for the prior-art statement is limited to the search range listed in §7.2.** What was
   done is the general web search with the five queries and the complete citation list of Orel 2015
   (6 works, down to title and abstract; none touches Hamiltonicity). **MathSciNet and zbMATH could not
   be reached, and the arXiv full-text search, the scanning of tables of contents of proceedings and the
   like, and non-English literature are not done.** For the citation list, too, what was obtained goes
   only as far as the abstracts, not the bodies.
   Therefore this document does not claim "world first". What it claims goes only as far as
   "**within the search range of this ledger it was not found**".
   **This single point is the weakest part of this document.**
6. **Whether it may be reported is the operator's decision.** Whether it goes outside, and if so in what
   range and how it should be worded, is decided neither by this working session nor by this ledger.

---

## 10. Attribution table

| What | Who | When / where |
|---|---|---|
| The definition of the graph HGLₙ(F₄), the cardinality formula, the degree formula (Prop. 12), Open Problem 16 | **Marko Orel** | *On generalizations of the Petersen graph and the Coxeter graph*, **Electron. J. Combin. 22(4) (2015), #P4.27** (Published Nov 13, 2015) |
| The Hamilton cycles in HGL₃(F₄), SGL₄(F₂), SGL₅(F₂) | **Marko Orel** (by the Concorde TSP Solver [5]) | Same as above, corresponding to p.14 |
| The enumeration underlying the cardinality formula | **[20, Corollary 5.32]** (Orel's citation) / [2, Theorem 9.5.7] (Brouwer–Cohen–Neumaier) / [6, Lemma 2.12] | Orel's bibliography |
| The cardinality of the hermitian variety used in the degree count | **[3, Theorem 8.1]** (Orel's citation) | Same as above |
| The search method for Hamilton cycles (rotation plus extension) | **L. Pósa** (1976) | Classical |
| The survey of Hamiltonicity of vertex-transitive graphs (the 5 known exceptions) | Orel's citation **[9]** (survey) | Same as above |
| **The cycle in HGL₄(F₄) itself** | **A working record** (a working session, 2026-09-07) | `hgl4_cycle.npy` |
| **The independent verifier (numpy version) and the certificate** | The same **working record** (a working session) | `r909_verify.py` / `r909_hgl4_certificate.md` |
| **The standard-library-only verifier, the delivery format, and this document** | The same **working record** (a working session) | `verify_hgl4.py` / `hgl4_cycle.txt` / this document |
| The correction "the families are just the two, HGLₙ(F₄) and SGLₙ(F₂)" | The same **working record** (self-correction N61-1) | Open Problem 16 of the printed paper |

**The reference numbers** ([2][3][5][6][9][20]) are those of the bibliography of Orel 2015, and this
document **does not copy that bibliography**. To check which works the numbers point to, consult the
printed paper.

---

**Notes.** The numbered markers in the text gloss names that are internal to the working side's ledger. The wording of the text itself is left as it is.

[^1]: A name internal to the working side's ledger: a "working record" is the record of one unit of work.

[^2]: A name internal to the working side's ledger: a "working session" is the unit into which the work is divided.

[^3]: A name internal to the working side's ledger: the "ledger" is the bound collection of the working records.

[^4]: A name internal to the working side's ledger: a "record number" (written with a ★ in the ledger) is the ledger's own numbering; it has no bearing on priority in the literature.

[^5]: A name internal to the working side's ledger: "N61-1" is the tag number of a point raised at examination.

[^6]: A name internal to the working side's ledger: "Wall N" numbers a rule of examination that the working side imposes on itself. Wall 19 is "do not treat 'not found by the search' as evidence that there is none — little attention is not evidence of difficulty".

[^7]: A name internal to the working side's ledger: "K59-W1" is the tag number of a point raised at examination, entered as a candidate for such a rule (a "wall candidate").