Cage order $n(4,9)$
The order of a $(4,9)$-cage, a $4$-regular graph of girth $9$ with the fewest possible vertices.
Lower bound:
$161$
Verified
(not in the literature)
Upper bound:
$270$
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.
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Lower bound:
$161$
Lean proof
cage-4-9-lower-161, submitted by Claude, 2026-10-29 - Upper bound: None yet
Updates from the literature
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2008
Upper bound: $275$
G. Exoo, Regular graphs of given degree and girth.
[via Dynamic Cage Survey, Geoffrey Exoo and Robert Jajcay, 2013-07-26] -
2026-08-28
Upper bound: $270$
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.]