{"id":51005250,"url":"https://github.com/chenle02/edge-3-critical-graphs-data","last_synced_at":"2026-06-20T20:01:26.738Z","repository":{"id":365869376,"uuid":"1265660144","full_name":"chenle02/edge-3-critical-graphs-data","owner":"chenle02","description":"Census data, audit reports, and code for: Chen \u0026 Shan, Exploring the world of edge-chromatic 3-critical 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Data and code for \"Exploring the world of edge-chromatic 3-critical graphs\"\n\n[![License: MIT](https://img.shields.io/badge/License-MIT-yellow.svg)](https://opensource.org/licenses/MIT)\n[![GitHub release](https://img.shields.io/github/v/release/chenle02/edge-3-critical-graphs-data)](https://github.com/chenle02/edge-3-critical-graphs-data/releases)\n[![verify](https://github.com/chenle02/edge-3-critical-graphs-data/actions/workflows/verify.yml/badge.svg)](https://github.com/chenle02/edge-3-critical-graphs-data/actions/workflows/verify.yml)\n[![website](https://img.shields.io/badge/website-online-blue)](https://chenle02.github.io/edge-3-critical-graphs-data/)\n\u003c!-- [![DOI](https://zenodo.org/badge/DOI/10.5281/zenodo.TODO.svg)](https://doi.org/10.5281/zenodo.TODO) --\u003e\n\n![A sample of nontrivial edge-chromatic 3-critical graphs (orange = degree-2 vertices)](assets/hero.png)\n\n**Website \u0026 interactive graph explorer:** \u003chttps://chenle02.github.io/edge-3-critical-graphs-data/\u003e\n\nThis repository archives the census data, machine-readable audit reports, and\nsearch/audit code referenced in the paper\n\n\u003e Le Chen and Songling Shan,\n\u003e *Exploring the world of edge-chromatic 3-critical graphs*, 2026.\n\nA graph with maximum degree 3 is (edge-chromatic) **3-critical** if it is\nconnected, has chromatic index 4, and deleting any edge lowers the chromatic\nindex to 3.  A 3-critical graph is **nontrivial** if it contains no\n3-overfull subgraph.  The paper lists the numbers of nontrivial\n3-critical graphs for all orders through 22, records complete computer\ncensuses at orders 19 and 21, reproduces the known order-22 count, gives a\nfinite construction ledger for the audited census, and proves two structural\nresults delimiting the cyclic 3-edge-cut method (an all-order one-sided\ntriangle-cap reduction, and an impossibility theorem for the\nboundary-completion reconstruction, with explicit witnesses of orders 25\nand 27).\n\n## Quickstart\n\n```python\nimport json, gzip, networkx as nx\n\n# load a census file (.json, or gzip.open(..., \"rt\") for .json.gz)\ndata = json.load(open(\"results/order_13_delta_3.json\"))\nprint(data[\"survivor_count\"], \"nontrivial survivors at order\", data[\"order\"])\n\n# rebuild any survivor as a networkx graph from its edge list\nG = nx.Graph(); G.add_edges_from(data[\"survivors\"][0][\"edges\"])\nprint(\"example survivor:\", G)\n```\n\nBrowse every order interactively (live graph rendering) on the\n[website](https://chenle02.github.io/edge-3-critical-graphs-data/).\n\n## Layout\n\n```\nresults/   complete census output files, orders 4 through 22\nreports/   machine-readable audit reports cited in the paper\ncode/      search pipeline and audit scripts\n```\n\n## Census files (`results/`)\n\nEach `order_n_delta_3.json` (or `.json.gz`) records the full census run at\norder `n`: number of generated 2-connected subcubic graphs, number of\n3-critical graphs, number of overfull ones, and the list of nontrivial\nsurvivors in graph6 format.  SHA-256 hashes of the files backing the paper's\nheadline counts:\n\n| File | Survivors | SHA-256 |\n|---|---:|---|\n| `order_13_delta_3.json` | 14 | `799aae0712bfb53b10279cdb178abd9ae2b55924f134d2fd3f9154ad4527ef10` |\n| `order_15_delta_3.json` | 94 | `c5c391d32a4019a6765e236ac9bfbd292572722073fbb72312d4ecbf91293162` |\n| `order_17_delta_3.json` | 774 | `c7447e9626f53cdb8a381376ec76eb95a58368bab85dd16dd02ba6d4f7b9a269` |\n| `order_19_delta_3.json` | 6,984 | `9f4eff7e13636fce3bcd3fc69cd2a3dfb36f1ad80e01656509fc6b9927f92b1e` |\n| `order_20_delta_3.json.gz` | 0 | `08e25a09ecdedd06b8904342203307ad71728aa609f5469d160ac9ce32ac8ed0` |\n| `order_21_delta_3.json.gz` | 70,530 | `2f9c3e46d744dc0b62a95631e050af657806ec76ae03a286f7e845c69cff24db` |\n| `order_22_delta_3.json.gz` | 1 | `57dbfccd9cb352564f5422530c9a0b7e269148c9789bf040f0dfd7ab96ed553e` |\n\nThe hashes for `order_20`, `order_21`, and `order_22` are of the compressed files, as\narchived here and as recorded during the original runs.\n\n## Audit reports (`reports/`)\n\nReports are archived exactly as produced by the audit runs (provenance\ncopies; JSON is the canonical record, Markdown siblings are human-readable\nsummaries).  The reports cited in the paper:\n\n| Paper reference | File(s) |\n|---|---|\n| Finite boundary-completion audit, orders 13-19 | `lemma24_boundary_completion_census_audit_20260601_190638.json.gz` |\n| Finite boundary-completion audit, order 21 | `lemma24_boundary_completion_census_audit_20260601_212852.json.gz` |\n| Exhaustive tight-side enumeration, orders 7-13 | `lemma_e_stress_test_hereditary_failing_sides_20260610_045602.json` |\n| Exhaustive tight-side enumeration, orders 15-17 | `lemma_e_stress_test_hereditary_failing_sides_20260610_081126.json` |\n| Gluing search and witness list (92 witnesses, orders 25/27) | `lemma_e_phase2_ambient_embedding_hunt_20260610_120153.json` |\n| Order-13 construction classification | `order13_triangle_blowup_classification.json`, `delta3_blowup_chain_9_11_13.json` |\n| Order-15/17 construction passes | `songling_order15_order17_generation_verification.json`, `..._second_pass.json`, `..._third_pass_hajos.json` |\n| Residual records | `songling_remaining_residue_dossier.json` |\n| Snark-deletion comparisons | `songling_snark_critical_subgraph_audit.json`, `songling_order17_snark_candidate_audit.json`, `songling_sve_hajos_followup_audit_20260505.*` |\n| Cyclic 3-cut side characterization ledger | `songling_h_characterization_ledger_20260602_ledger.json` |\n\nNote on terminology: some artifacts use the internal working name\n\"Lemma E\" for the all-order boundary-completion property; that property is\nexactly the one refuted by the impossibility theorem of the paper.\n\n## Code (`code/`)\n\n- `code/critical_graph_search/` and `code/main.py`: the census pipeline\n  (graph generation via `geng` from the nauty suite, pruning filters,\n  bitmask backtracking edge-coloring, criticality and overfull tests).\n- `code/scripts/audit_lemma24_boundary_completion_repair.py`: the\n  boundary-completion audit behind the finite audit proposition.\n- `code/scripts/audit_songling_cyclic3_kempe_chain_request.py`: shared\n  cyclic 3-cut side-enumeration helpers.\n- `code/scripts/lemma_e_stress_test_hereditary_failing_sides_20260610.py`:\n  exhaustive tight-side-shape enumeration (orders 7-17).\n- `code/scripts/lemma_e_phase2_ambient_embedding_hunt_20260610.py`:\n  the bounded gluing search producing the 92 witnesses.\n- `code/scripts/independent_verify_order25_witness.py`: an independent\n  verifier for the order-25 witness of the impossibility theorem.  Its\n  3-edge-coloring solver and small-cut scans are written from scratch; the\n  overfull test cross-checks the census-validated exhaustive enumerator\n  `critical_graph_search.density_filter.has_overfull_subgraph` against an\n  independent 2,000,000-sample random odd-subset spot-check and a cut-form\n  arithmetic argument.  It requires `networkx` and `numpy` plus the bundled\n  `critical_graph_search` package shipped in `code/`.\n\nThe audit scripts are archived as run; some refer to paths in the private\nresearch repository where they were executed.  The independent verifier\nruns against this repository alone and reproduces, in under a minute, every\nproperty of the order-25 witness claimed in the paper.\n\nRequirements: Python 3.10+, `networkx`, and `numpy` (the verifier also\nimports the bundled `critical_graph_search` package shipped in `code/`);\ngraph generation additionally requires `geng` from the nauty suite.\n\n## The order-25 witness\n\nThe impossibility theorem's witness is the order-25 graph with graph6 string\n\n```\nX???C@?K@OOae?DOGP@D?QO?C????G??G??A?G?G??A_??P?_?@\n```\n\nRun `python3 code/scripts/independent_verify_order25_witness.py` to check\nall of its claimed properties end to end.\n\n## Acknowledgment\n\nThis work was completed in part with resources provided by the Auburn University Easley Cluster.\n\n## License\n\nMIT (see LICENSE).\n","project_url":"https://awesome.ecosyste.ms/api/v1/projects/github.com%2Fchenle02%2Fedge-3-critical-graphs-data","html_url":"https://awesome.ecosyste.ms/projects/github.com%2Fchenle02%2Fedge-3-critical-graphs-data","lists_url":"https://awesome.ecosyste.ms/api/v1/projects/github.com%2Fchenle02%2Fedge-3-critical-graphs-data/lists"}