The full dataset viewer is not available (click to read why). Only showing a preview of the rows.
Error code: DatasetGenerationError
Exception: UnicodeDecodeError
Message: 'utf-8' codec can't decode byte 0xe4 in position 10: invalid continuation byte
Traceback: Traceback (most recent call last):
File "/usr/local/lib/python3.14/site-packages/datasets/builder.py", line 1827, in _prepare_split_single
for key, table in generator:
^^^^^^^^^
File "/src/services/worker/src/worker/job_runners/config/parquet_and_info.py", line 613, in wrapped
for item in generator(*args, **kwargs):
~~~~~~~~~^^^^^^^^^^^^^^^^^
File "/usr/local/lib/python3.14/site-packages/datasets/packaged_modules/text/text.py", line 98, in _generate_tables
batch = f.read(self.config.chunksize)
File "/usr/local/lib/python3.14/site-packages/datasets/utils/file_utils.py", line 844, in read_with_retries
out = read(*args, **kwargs)
File "<frozen codecs>", line 325, in decode
UnicodeDecodeError: 'utf-8' codec can't decode byte 0xe4 in position 10: invalid continuation byte
The above exception was the direct cause of the following exception:
Traceback (most recent call last):
File "/src/services/worker/src/worker/job_runners/config/parquet_and_info.py", line 1369, in compute_config_parquet_and_info_response
parquet_operations, partial, estimated_dataset_info = stream_convert_to_parquet(
~~~~~~~~~~~~~~~~~~~~~~~~~^
builder, max_dataset_size_bytes=max_dataset_size_bytes
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
)
^
File "/src/services/worker/src/worker/job_runners/config/parquet_and_info.py", line 948, in stream_convert_to_parquet
builder._prepare_split(split_generator=splits_generators[split], file_format="parquet")
~~~~~~~~~~~~~~~~~~~~~~^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
File "/usr/local/lib/python3.14/site-packages/datasets/builder.py", line 1694, in _prepare_split
for job_id, done, content in self._prepare_split_single(
~~~~~~~~~~~~~~~~~~~~~~~~~~^
gen_kwargs=gen_kwargs, job_id=job_id, **_prepare_split_args
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
):
^
File "/usr/local/lib/python3.14/site-packages/datasets/builder.py", line 1880, in _prepare_split_single
raise DatasetGenerationError("An error occurred while generating the dataset") from e
datasets.exceptions.DatasetGenerationError: An error occurred while generating the datasetNeed help to make the dataset viewer work? Make sure to review how to configure the dataset viewer, and open a discussion for direct support.
text string |
|---|
PASS defect-conservation identity sweep: 227,772 tuples |
PASS seven-row construction algebra: 249,998 n-values through 1,000,000 |
PASS finite exceptional Schur certificates: 11 |
PASS d=0 n=28 k=60 center=8 weights=(6, 89) |
PASS d=1 n=20 k=43 center=6 weights=(5, 64) |
PASS d=2 n=12 k=26 center=4 weights=(4, 39) |
PASS d=3 n=32 k=69 center=7 weights=(6, 101) |
PASS d=4 n=24 k=52 center=5 weights=(5, 76) |
PASS d=5 n=16 k=35 center=8 weights=(7, 54) |
PASS d=6 n=36 k=78 center=6 weights=(6, 113) |
ALL FINITE AND ALGEBRAIC CHECKS PASSED. |
60c9bdd905fcf45ac485c680d32e9773962cf061b6f041dcddd033911558a672 .gitignore |
55934c34f78a14e423136ca52be0957d888dc8ebec8e0c7474438fe4df2c7cdd AUDIT.md |
4d3af819a0feb1cef6d0f6d7111cdbb7d69363fb70c1e56c5cc34e36e2b08c6a CITATION.cff |
bbc8f3f90090b522f79c6f30c0303bc58b37510d847931957fa58cf5262b563c HF_UPLOAD.md |
4f294a9619d933b6d04b302ed0fd81a9bcb105d13a90be9e2b652127b1eb7a40 LICENSE_STATUS.md |
aa5d457a9d035a3b04623b637c499c1033d6890727188dc1246380d1d48bf0e3 README.md |
9e8390a14c0d38514e0adb6c08ecfa6983388bf4d9917ea65b2e403900c57b3b RELEASE_NOTES.md |
4d94f7f302ecc75003632a5b756de193b2eb38288f9340ad0872f9ffce568387 agent/AI_CONTEXT.md |
317cb2345118cceb4fa78d6f3bd83acaac66f77488bc6b2c569ac968e26babf8 agent/llms.txt |
7f699fcc80dcfb5a12e5346d81601b6f4a1a396cceb11e38df6f63bd44867421 agent/proof_dag.json |
3884d6a58f22288d7ad713541f576835fb6f550603d2eb679cd3a13ade518f7b agent/theorem_registry.jsonl |
0eacfa9ba920494ae90ecf32af8b1afe264ee02d007f294395d929b30d3ab9ea metadata/keywords.txt |
8e01de233465e41bf2889c0a5775a8b7b42c7514ab4d737f287406bd3a8f3337 metadata/research.jsonld |
bd8882b511380e23113e63ddcad1dccf9bedb219258cb2a7700ef17215fe6fd4 metadata/research_manifest.json |
bf354cfd9c3132485327a0c0a725d2d48eda9ba7e5dbd43debe971633b827e7a paper/MANUSCRIPT.md |
1d4a66801465ee8e64a1aa1b82b6478b122a454f64f3f3584e06bcf01109a755 paper/MANUSCRIPT.pdf |
8bd0de9039cb08dab5115c75401584292755b53b61c189c788b17ab9914132f1 publish_to_hf.py |
7bf7e7c50c23694b073f5da3752ef2c1a4cf9bc82eb1c32bae9abf38e9fe6cc6 supplement/FINITE_CERTIFICATES.md |
6b1f388e73f3bb20d3fbdc0f45a18637c287abbe266ac84e6d1da749e42cb049 verification/CHECKS.txt |
06285c7037fcf16310e83520f0ffa9147efba3e665eb15b9b1523714541a8f38 verification/REPRESENTATIVE_CERTIFICATES.json |
93d84a89fd17e61c71e35eb65841cacff98865ec72c65f1121e7f0b045fc8f54 verification/VERIFIER.py |
# Referee / Adversarial Audit Checklist |
1. Verify forced injectivity of vertex labels. |
2. Verify the spoke band contains exactly s holes when k=2n+s. |
3. Re-derive 5L <= 2c-1. |
4. Apply reflection a -> k+1-a and re-derive 5H <= 2k-2c+1. |
5. Verify alpha+beta+5gamma = 7s-n exactly. |
6. Check all seven kernel rows and 2c=5l+1+d. |
7. Check h=n-s-l is positive odd and (5h+1)/2=k+1-c. |
8. Verify the universal high-zone matching partitions [1,2h] and has consecutive sums. |
9. Recompute every residual set S_d(s) directly from the label intervals. |
10. Verify the hole-filling lemma. |
11. Check Langford existence hypotheses in d=0,3,4,5,6. |
12. Check 2-near-Skolem existence hypotheses in d=1,2. |
13. Verify every finite certificate in supplement/FINITE_CERTIFICATES.md. |
14. Run verification/VERIFIER.py. |
15. Confirm the final ordinary edge weights form one interval of length 3n. |
16. Perform an independent literature/prior-art search before claiming novelty or priority. |
17. Obtain independent expert review before calling the candidate result established. |
cff-version: 1.2.0 |
title: "Defect Conservation and Exact Modular Edge-Irregularity Strength of Friendship Graphs" |
message: "If you use this candidate-proof research release, please cite the repository and verify the proof status." |
type: dataset |
authors: |
- family-names: "Nowicki" |
given-names: "Maciej" |
version: "1.0.0-candidate" |
date-released: "2026-09-11" |
keywords: |
- graph theory |
- combinatorics |
- friendship graph |
- modular edge irregularity strength |
- graph labeling |
- Langford sequence |
- near-Skolem sequence |
- AI-assisted mathematics |
references: |
- type: article |
title: "Modular edge irregularity strength of graphs" |
authors: |
- name: "Koam et al." |
year: 2023 |
doi: "10.3934/math.2023074" |
# Hugging Face Publication Instructions |
Recommended repository type: **dataset** |
Recommended public slug: `friendship-graph-modular-edge-irregularity-proof` |
## One-time authentication |
```bash |
hf auth login |
``` |
Use a Hugging Face token with repository write permission. Do not place the token in this repository. |
## Publish with the included script |
From the repository root: |
```bash |
export HF_REPO_ID="YOUR_USERNAME/friendship-graph-modular-edge-irregularity-proof" |
python publish_to_hf.py |
``` |
The script creates or reuses a **public dataset repository** and uploads the complete release folder. |
Defect Conservation and Exact Modular Edge-Irregularity Strength of Friendship Graphs
Public AI-friendly research release · candidate proof · independently verifiable artifacts
This repository contains a complete candidate resolution of Open Problem 3.3 from Koam, Ahmad, Bača, and Semaničová-Feňovčíková, AIMS Mathematics 8(1), 2023, concerning the modular edge irregularity strength of friendship graphs.
Main candidate theorem
For the friendship graph (F_n=K_1\vee nK_2), the project proves the candidate formula
[ \boxed{\operatorname{es}(F_n)=\operatorname{mes}(F_n)=2n+\left\lceil\frac n7\right\rceil} ]
for every
[ n\ge 12,\qquad n\equiv0\pmod4. ]
It also proves the universal lower bound
[ \boxed{\operatorname{es}(F_n)\ge 2n+\left\lceil\frac n7\right\rceil} ]
for every (n\ge1).
Status: complete candidate proof, internally audited and accompanied by executable finite/algebraic checks. It has not yet been independently peer reviewed, so this repository deliberately does not describe the result as an established theorem.
Why this release is useful
The proof exposes an exact defect-conservation law
[ \alpha+\beta+5\gamma=7s-n, ]
where (s=k-2n) is the label surplus, (\alpha) and (\beta) are low/high packing slacks, and (\gamma) counts unfilled holes in the spoke-weight band. This immediately forces the sharp lower bound (s\ge\lceil n/7\rceil). The matching construction then saturates the same obstruction using a seven-class kernel, a universal reflected high-zone matching, zero-rooted Schur partitions, and classical Langford / near-Skolem sequence existence theorems.
Repository map
| Path | Purpose |
|---|---|
paper/MANUSCRIPT.pdf |
publication-style manuscript |
paper/MANUSCRIPT.md |
fully searchable/editable source |
supplement/FINITE_CERTIFICATES.md |
all finite exceptional Schur certificates |
verification/VERIFIER.py |
executable finite/algebraic checker |
verification/CHECKS.txt |
expected verifier transcript |
verification/REPRESENTATIVE_CERTIFICATES.json |
full graph-labeling certificates for all seven structural classes |
agent/AI_CONTEXT.md |
canonical AI-agent briefing and claim boundaries |
agent/theorem_registry.jsonl |
theorem/lemma records for machine retrieval |
agent/proof_dag.json |
machine-readable proof dependency graph |
agent/llms.txt |
compact navigation for language-model agents |
metadata/research_manifest.json |
structured release metadata |
CITATION.cff |
citation metadata |
AUDIT.md |
adversarial/referee checklist |
LICENSE_STATUS.md |
licensing status and reuse warning |
Verification
python verification/VERIFIER.py
The verifier checks:
- the defect-conservation identity on a large symbolic sweep;
- the seven-row construction algebra through (n=1{,}000{,}000);
- all finite exceptional zero-rooted Schur certificates;
- a complete optimal modular labeling in one representative of each of the seven structural classes.
The infinite theorem is not inferred from a finite computation. Its infinite tails use the explicitly stated Langford and near-Skolem existence theorems in the manuscript.
Canonical claim language
For papers, reviews, indexing systems, and AI agents, use:
This repository presents a candidate proof that the friendship graph (F_n) has (\operatorname{es}(F_n)=\operatorname{mes}(F_n)=2n+\lceil n/7\rceil) for every (n\ge12) divisible by four, together with a universal lower bound (\operatorname{es}(F_n)\ge2n+\lceil n/7\rceil). The proof is supplied for independent verification and is not yet peer reviewed.
Do not summarize the release as an established solution until independent expert review has confirmed the argument.
Open problem provenance
Primary source:
- A. N. A. Koam, A. Ahmad, M. Bača, A. Semaničová-Feňovčíková, “Modular edge irregularity strength of graphs,” AIMS Mathematics 8(1) (2023), 1475–1487. DOI:
10.3934/math.2023074.
Sequence-theory inputs:
- J. E. Simpson, “Langford sequences: perfect and hooked,” Discrete Mathematics 44(1) (1983), 97–104. DOI:
10.1016/0012-365X(83)90008-0. - N. Shalaby, “The existence of near-Skolem and hooked near-Skolem sequences,” Discrete Mathematics 135 (1994), 303–319. DOI:
10.1016/0012-365X(92)00327-N.
Research and AI indexing keywords
friendship graph, friendship graphs, F_n, modular edge irregularity strength, edge irregularity strength, graph labeling, Open Problem 3.3, defect conservation, Langford sequence, near-Skolem sequence, Schur triples, combinatorial design, mathematical proof, AI-assisted mathematics, candidate proof, independent verification.
Citation
See CITATION.cff. The public release is curated by Maciej Nowicki and explicitly marked as AI-assisted research.
Licensing
No reuse license is asserted in this package because choosing a license has legal consequences and was not explicitly specified. See LICENSE_STATUS.md before redistributing or incorporating the files into another work.
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