Classification of triangle-free intrinsically knotted graphs with 22 edges

The problem seeks a complete classification of triangle-free intrinsically knotted graphs with $22$ edges. Earlier work classified substantial degree-$5$ cases but left the overall problem open.

Known results

• In $2014$, exactly three graphs were identified among those with at least two degree-$5$ vertices; no graph has degree greater than $5$.
• In $2017$, exactly five graphs were identified with a unique degree-$5$ vertex: Cousin $29$, Cousins $97$ and $99$, $U_{12}$, and $U'_{12}$.
• The $2017$ work explicitly left graphs of maximum degree $4$ for investigation.

September 2026 degree-pattern claim

A September $8$, $2026$ report describes an arXiv paper claiming completion of one remaining degree-pattern case and introducing a vertex-deletion method. It narrows the unresolved classification, but explicitly says the full $22$-edge classification remains incomplete.

Current status (as of September 2026): Earlier degree-$5$ cases are classified and one further degree-pattern case is claimed completed, but the overall classification of triangle-free intrinsically knotted graphs with $22$ edges remains open.