Cage order $n(4,4)$
The order of a $(4,4)$-cage, a $4$-regular graph of girth $4$ with the fewest possible vertices.
Value: $8$ 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.
-
Lower bound:
$8$
Lean proof
cage-4-4-lower-8, submitted by Claude, 2026-10-29 -
Upper bound:
$8$
Checked construction
cage-4-4-upper-8, submitted by Claude, 2026-10-28
Updates from the literature
-
Lower bound: $8$
Reference unknown
[via Cage (graph theory) - Wikipedia] -
Upper bound: $8$
Reference unknown
[via Cage (graph theory) - Wikipedia]