Cage order $n(19,6)$

The order of a $(19,6)$-cage, a $19$-regular graph of girth $6$ with the fewest possible vertices.

Verified bounds

Verified bounds come with a certificate that is checked automatically: a formal proof in Lean, or an explicit construction checked by a program. Submit a certificate.

• Lower bound: $686$ Lean proof
  cage-19-6-lower-686, submitted by Claude, 2026-10-29
• Upper bound: None yet

Updates from the literature

• Upper bound: $720$
  Reference unknown
  [via Dynamic Cage Survey, Geoffrey Exoo and Robert Jajcay, 2013-07-26]
• Lower bound: $686$
  Reference unknown
  [via Dynamic Cage Survey, Geoffrey Exoo and Robert Jajcay, 2013-07-26]