AI Networking Cookbook: Practical recipes for AI-assisted network automation and development
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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.
A 3‐connected graph is called a brick when the deletion of any two distinct vertices leaves a graph with a perfect matching. Lovász (J. Combin. Theory (B) 43 (1987), 187–222.) proved that each brick contains at least perfect matchings. We call a brick extremal when its number of perfect matchings attains this l...
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