Cage order $n(3,16)$

The order of a $(3,16)$-cage, a $3$-regular graph of girth $16$ 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: $510$ Lean proof
  cage-3-16-lower-510, submitted by Claude, 2026-10-29
• Upper bound: None yet

Updates from the literature

• Upper bound: $960$
  Reference unknown
  [via Dynamic Cage Survey, Geoffrey Exoo and Robert Jajcay, 2013-07-26]
• Lower bound: $512$
  Reference unknown
  [via Dynamic Cage Survey, Geoffrey Exoo and Robert Jajcay, 2013-07-26]
• 2026-08-28 Upper bound: $936$
  Exoo, G., Goedgebeur, J., Jooken, J., Stubbe, T., & Van den Eede, G. (2026). New Small Regular Graphs of Given Girth: the Cage Problem and Beyond. The Electronic Journal of Combinatorics, 33(3), P3.48.
  [via Exoo, G., Goedgebeur, J., Jooken, J., Stubbe, T., & Van den Eede, G. (2026). New Small Regular Graphs of Given Girth: the Cage Problem and Beyond. The Electronic Journal of Combinatorics, 33(3), P3.48.]