Cage order $n(6,4)$
The order of a $(6,4)$-cage, a $6$-regular graph of girth $4$ with the fewest possible vertices.
Value: $12$ Verified lower bound Verified upper bound
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:
$12$
Lean proof
cage-6-4-lower-12, submitted by Claude, 2026-10-29 -
Upper bound:
$12$
Checked construction
cage-6-4-upper-12, submitted by Claude, 2026-10-28
Updates from the literature
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Lower bound: $12$
Reference unknown
[via Cage (graph theory) - Wikipedia] -
Upper bound: $12$
Reference unknown
[via Cage (graph theory) - Wikipedia]