Canonical forms for graphs in time n^{O(log n)}
Year: 2026
Type: preprint
Abstract: We describe a deterministic algorithm that, given a graph X on n vertices, computes a canonical form Can(X), an isomorphism X → Can(X), and generators of Aut(X) in time exp(O(log² n)) = n^{O(log n)}. More generally, for a permutation group G ≤ Sym(Ω) given by generators, it computes a G-canonical image of a string x: Ω → Σ, a transporter, and the full stabilizer in time exp(O(log² |Ω|)) times a polynomial in the input description. The algorithm converts the recent deterministic n^{O(log n)} isomorphism algorithm of Hajiaghayi and JafariRaviz, which outputs isomorphisms and automorphism groups but not canonical forms, into a canonization algorithm. To our knowledge, it improves the best previous canonization bound, Babai's exp((log n)^{O(1)}) of 2019, to the new isomorphism bound. The new ingredients are canonical codes for constant-size local endpoints, built by canonizing successive invariant windows under the stabilizer of a common canonical prefix; a canonical row terminal, using soluble implicit multicoset canonization and affine canonization under a soluble permutation cover with deterministic root selection; a covariant overcoset lemma that lets a branch forget intermediate restrictions while keeping full automorphism groups; and a parity-correct extension of rectangle placements. The result is conditional on the theorems of Hajiaghayi and JafariRaviz that it imports. Provenance and status. The proofs were produced by an OpenAI GPT model, in an AI-assisted research program designed and directed by Matthew Liston. Claude (Anthropic), which coordinated the program, checked selected lemmas; those checks were careful reads, not referee reports. The proof record was audited by an independent AI referee (an OpenAI model, run in a session separate from the one that produced the proof), which checked every interface the record imports from Hajiaghayi and JafariRaviz, and the results it uses from Schweitzer–Wiebking and Luks–Miyazaki, against those papers. Its verdict was "proved with fixable gaps", conditional on the imported Hajiaghayi–JafariRaviz results. It found no fatal or major issue, and the six minor repairs it requested are incorporated in the revised record on which the paper is based (Appendix A of the paper lists the repairs, and Appendix B reproduces the revised record). The referee did not re-referee the Hajiaghayi–JafariRaviz paper. There has been no human refereeing yet, and there is no machine-checked formal proof of the conversion. The trusted statement of the Lean formalization of Hajiaghayi and JafariRaviz is their main theorem, for graphs; it does not include the results imported here, or this conversion. We are confident that the argument is correct and intend to verify it formally; the program's formal-verification capacity is currently committed to other results, so this version is released now. Files. graph-canonization-nlogn-v1.pdf: the paper (47 pages; Appendix B reproduces the revised proof record verbatim). SHA-256 ecaa5d1ed89684ceb4902e7618bad6735770e174453610e3d1db1dd27ee69b1f graph-canonization-nlogn-v1.zip: the paper and its LaTeX source; the revised proof record (version 2) and the original record as audited; the AI referee report; README and SHA256SUMS. SHA-256 151d572e066e817b9b1c7fd3f7df576dba8034e3b9376467d245e846022edcc4 Records inside the archive: proofs/Graph_Canonization_Log2_Proof_v2_2026-11-06.txt (revised record, version 2): SHA-256 14dbc769e3c2712a7b97f25c0691ae30e05412af6a9a32e4f04a27f5e14296b0 proofs/Graph_Canonization_Log2_Proof_2026-11-06.txt (original record, as audited): SHA-256 feae4a9dc42449e7b65a9e2a6515b30d4b5801caa89ffdbb77fb41369fe69796 ai-referee-reports/Graph_Canonization_Log2_Referee_Report_2026-11-06.txt (AI referee report): SHA-256 0a79b052866348fbf73dafb15ef666cb012607f28705af6e3c3e32ad01d4d7e4 Contact: [email protected]
Author: Liston, Matthew
Language: English
Topic: Finite Group Theory Research
Subfield: Discrete Mathematics and Combinatorics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7225419261