https://github.com/chenle02/edge-3-critical-graphs-data
Census data, audit reports, and code for: Chen & Shan, Exploring the world of edge-chromatic 3-critical graphs
https://github.com/chenle02/edge-3-critical-graphs-data
census chromatic-index combinatorics critical-graphs edge-coloring graph-theory nauty reproducible-research snarks
Last synced: about 1 month ago
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Census data, audit reports, and code for: Chen & Shan, Exploring the world of edge-chromatic 3-critical graphs
- Host: GitHub
- URL: https://github.com/chenle02/edge-3-critical-graphs-data
- Owner: chenle02
- License: mit
- Created: 2026-06-11T01:19:43.000Z (about 1 month ago)
- Default Branch: master
- Last Pushed: 2026-06-19T08:28:33.000Z (about 1 month ago)
- Last Synced: 2026-06-19T09:29:01.322Z (about 1 month ago)
- Topics: census, chromatic-index, combinatorics, critical-graphs, edge-coloring, graph-theory, nauty, reproducible-research, snarks
- Language: Python
- Size: 5.22 MB
- Stars: 0
- Watchers: 0
- Forks: 0
- Open Issues: 0
-
Metadata Files:
- Readme: README.md
- License: LICENSE
- Citation: CITATION.cff
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README
# Data and code for "Exploring the world of edge-chromatic 3-critical graphs"
[](https://opensource.org/licenses/MIT)
[](https://github.com/chenle02/edge-3-critical-graphs-data/releases)
[](https://github.com/chenle02/edge-3-critical-graphs-data/actions/workflows/verify.yml)
[](https://chenle02.github.io/edge-3-critical-graphs-data/)
**Website & interactive graph explorer:**
This repository archives the census data, machine-readable audit reports, and
search/audit code referenced in the paper
> Le Chen and Songling Shan,
> *Exploring the world of edge-chromatic 3-critical graphs*, 2026.
A graph with maximum degree 3 is (edge-chromatic) **3-critical** if it is
connected, has chromatic index 4, and deleting any edge lowers the chromatic
index to 3. A 3-critical graph is **nontrivial** if it contains no
3-overfull subgraph. The paper lists the numbers of nontrivial
3-critical graphs for all orders through 22, records complete computer
censuses at orders 19 and 21, reproduces the known order-22 count, gives a
finite construction ledger for the audited census, and proves two structural
results delimiting the cyclic 3-edge-cut method (an all-order one-sided
triangle-cap reduction, and an impossibility theorem for the
boundary-completion reconstruction, with explicit witnesses of orders 25
and 27).
## Quickstart
```python
import json, gzip, networkx as nx
# load a census file (.json, or gzip.open(..., "rt") for .json.gz)
data = json.load(open("results/order_13_delta_3.json"))
print(data["survivor_count"], "nontrivial survivors at order", data["order"])
# rebuild any survivor as a networkx graph from its edge list
G = nx.Graph(); G.add_edges_from(data["survivors"][0]["edges"])
print("example survivor:", G)
```
Browse every order interactively (live graph rendering) on the
[website](https://chenle02.github.io/edge-3-critical-graphs-data/).
## Layout
```
results/ complete census output files, orders 4 through 22
reports/ machine-readable audit reports cited in the paper
code/ search pipeline and audit scripts
```
## Census files (`results/`)
Each `order_n_delta_3.json` (or `.json.gz`) records the full census run at
order `n`: number of generated 2-connected subcubic graphs, number of
3-critical graphs, number of overfull ones, and the list of nontrivial
survivors in graph6 format. SHA-256 hashes of the files backing the paper's
headline counts:
| File | Survivors | SHA-256 |
|---|---:|---|
| `order_13_delta_3.json` | 14 | `799aae0712bfb53b10279cdb178abd9ae2b55924f134d2fd3f9154ad4527ef10` |
| `order_15_delta_3.json` | 94 | `c5c391d32a4019a6765e236ac9bfbd292572722073fbb72312d4ecbf91293162` |
| `order_17_delta_3.json` | 774 | `c7447e9626f53cdb8a381376ec76eb95a58368bab85dd16dd02ba6d4f7b9a269` |
| `order_19_delta_3.json` | 6,984 | `9f4eff7e13636fce3bcd3fc69cd2a3dfb36f1ad80e01656509fc6b9927f92b1e` |
| `order_20_delta_3.json.gz` | 0 | `08e25a09ecdedd06b8904342203307ad71728aa609f5469d160ac9ce32ac8ed0` |
| `order_21_delta_3.json.gz` | 70,530 | `2f9c3e46d744dc0b62a95631e050af657806ec76ae03a286f7e845c69cff24db` |
| `order_22_delta_3.json.gz` | 1 | `57dbfccd9cb352564f5422530c9a0b7e269148c9789bf040f0dfd7ab96ed553e` |
The hashes for `order_20`, `order_21`, and `order_22` are of the compressed files, as
archived here and as recorded during the original runs.
## Audit reports (`reports/`)
Reports are archived exactly as produced by the audit runs (provenance
copies; JSON is the canonical record, Markdown siblings are human-readable
summaries). The reports cited in the paper:
| Paper reference | File(s) |
|---|---|
| Finite boundary-completion audit, orders 13-19 | `lemma24_boundary_completion_census_audit_20260601_190638.json.gz` |
| Finite boundary-completion audit, order 21 | `lemma24_boundary_completion_census_audit_20260601_212852.json.gz` |
| Exhaustive tight-side enumeration, orders 7-13 | `lemma_e_stress_test_hereditary_failing_sides_20260610_045602.json` |
| Exhaustive tight-side enumeration, orders 15-17 | `lemma_e_stress_test_hereditary_failing_sides_20260610_081126.json` |
| Gluing search and witness list (92 witnesses, orders 25/27) | `lemma_e_phase2_ambient_embedding_hunt_20260610_120153.json` |
| Order-13 construction classification | `order13_triangle_blowup_classification.json`, `delta3_blowup_chain_9_11_13.json` |
| Order-15/17 construction passes | `songling_order15_order17_generation_verification.json`, `..._second_pass.json`, `..._third_pass_hajos.json` |
| Residual records | `songling_remaining_residue_dossier.json` |
| Snark-deletion comparisons | `songling_snark_critical_subgraph_audit.json`, `songling_order17_snark_candidate_audit.json`, `songling_sve_hajos_followup_audit_20260505.*` |
| Cyclic 3-cut side characterization ledger | `songling_h_characterization_ledger_20260602_ledger.json` |
Note on terminology: some artifacts use the internal working name
"Lemma E" for the all-order boundary-completion property; that property is
exactly the one refuted by the impossibility theorem of the paper.
## Code (`code/`)
- `code/critical_graph_search/` and `code/main.py`: the census pipeline
(graph generation via `geng` from the nauty suite, pruning filters,
bitmask backtracking edge-coloring, criticality and overfull tests).
- `code/scripts/audit_lemma24_boundary_completion_repair.py`: the
boundary-completion audit behind the finite audit proposition.
- `code/scripts/audit_songling_cyclic3_kempe_chain_request.py`: shared
cyclic 3-cut side-enumeration helpers.
- `code/scripts/lemma_e_stress_test_hereditary_failing_sides_20260610.py`:
exhaustive tight-side-shape enumeration (orders 7-17).
- `code/scripts/lemma_e_phase2_ambient_embedding_hunt_20260610.py`:
the bounded gluing search producing the 92 witnesses.
- `code/scripts/independent_verify_order25_witness.py`: an independent
verifier for the order-25 witness of the impossibility theorem. Its
3-edge-coloring solver and small-cut scans are written from scratch; the
overfull test cross-checks the census-validated exhaustive enumerator
`critical_graph_search.density_filter.has_overfull_subgraph` against an
independent 2,000,000-sample random odd-subset spot-check and a cut-form
arithmetic argument. It requires `networkx` and `numpy` plus the bundled
`critical_graph_search` package shipped in `code/`.
The audit scripts are archived as run; some refer to paths in the private
research repository where they were executed. The independent verifier
runs against this repository alone and reproduces, in under a minute, every
property of the order-25 witness claimed in the paper.
Requirements: Python 3.10+, `networkx`, and `numpy` (the verifier also
imports the bundled `critical_graph_search` package shipped in `code/`);
graph generation additionally requires `geng` from the nauty suite.
## The order-25 witness
The impossibility theorem's witness is the order-25 graph with graph6 string
```
X???C@?K@OOae?DOGP@D?QO?C????G??G??A?G?G??A_??P?_?@
```
Run `python3 code/scripts/independent_verify_order25_witness.py` to check
all of its claimed properties end to end.
## Acknowledgment
This work was completed in part with resources provided by the Auburn University Easley Cluster.
## License
MIT (see LICENSE).