The biwheel conjecture for braces

Abstract

Let $b(n)$ denote the minimum number of perfect matchings in a simple brace of order $n$. De~Carvalho, Lucchesi and Murty (SIAM Journal on Discrete Mathematics, 27(2): (2013) 940-958) proved that $b(n)\ge(n-2)^2/8$ and conjectured that $b(n)=(n-2)^2/4$ for all sufficiently large even $n$. We prove this conjecture.