Breakthrough Tracker record
Six-author analysis tightens the Petersen Coloring counterexample to 40–52 vertices
Putman’s 112-vertex disproof remains independently supported. Goedgebeur, Jooken, Máčajová, Mattiolo, Mazzuoccolo and Ulyanov now give two 52-vertex counterexamples, a purely theoretical proof, an infinite family for every even order at least 60 and a computational lower bound of 40 vertices.
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- Stable ID
math-petersen-coloring-counterexample-2026- Revision
math-petersen-coloring-counterexample-2026.v3- Field
- Mathematics · Graph theory and computer-assisted proof
- Evidence
- Tier 1 · Peer reviewed: No
- Record state
- Provisional · Counterexample independently corroborated
- Last checked
AI role
Putman reports extensive OpenAI-model use in discovery, computational search, verification and preparation. Jooken reports GPT-5.6 Sol assistance with presentation and proof rewriting while stating that he independently verified every mathematical statement and proof. The 52-vertex follow-up does not disclose a substantive AI role.
Record details
- Problem or result
- Jaeger’s Petersen Coloring Conjecture for bridgeless cubic graphs
- Authors
- Bryce Putman; Jan Goedgebeur, Jorik Jooken, Edita Máčajová, Davide Mattiolo, Giuseppe Mazzuoccolo and Nikolay Ulyanov
- Institutions
- Leiden University; KU Leuven; Ghent University; Comenius University; University of Modena and Reggio Emilia
- Result date
- 112-vertex manuscripts August 8–14; six-author v4 revised September 30, 2026
Why it matters
The original construction refuted the conjecture. The later theoretical analysis narrows the possible minimum to 40–52 vertices and supplies counterexamples at every even order from 60 onward.
Limits
The current six-author analysis is not peer reviewed. Its theoretical proof and computational lower bound materially strengthen the record, but the exact minimum between 40 and 52 vertices remains unknown.
Sources
- Primary: Bryce Putman, 112-vertex Petersen-coloring counterexample
- Artifacts: Frozen graph, SAT, DRAT and verification artifacts
- Independent: Goedgebeur et al., theoretical analysis, two 52-vertex examples and lower bound
- Primary: Goedgebeur et al., 52-vertex counterexample and infinite family
Correction and revision history
- 2026-08-22 — Added after primary-source, scope, status, AI-role and limitation review.
- 2026-09-01 — Added the presented 52-vertex counterexample and infinite cyclically four-edge-connected family; the smaller graph remains independently unchecked.
- 2026-10-02 — Updated the evolving arXiv source to six-author v4, added the theoretical proof and infinite family, and narrowed the minimum interval to 40–52 vertices.
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