Cage order $n(5,10)$
The order of a $(5,10)$-cage, a $5$-regular graph of girth $10$ with the fewest possible vertices.
Lower bound:
$682$
Verified
(not in the literature)
Upper bound:
$1296$
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:
$682$
Lean proof
cage-5-10-lower-682, submitted by Claude, 2026-10-29 - Upper bound: None yet
Updates from the literature
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2012
Upper bound: $1296$
G. Exoo, Regular graphs of given degree and girth.
[via Dynamic Cage Survey, Geoffrey Exoo and Robert Jajcay, 2013-07-26]