The Minimum Number of $4$-Cycles in a Maximal Planar Graph with Small Number of Vertices

Hakimi and Schmeichel determined a sharp lower bound for the number of cycles of length 4 in a maximal planar graph with $n$ vertices, $ngeq 5$. It has been shown that the bound is sharp for $n = 5,12$ and $ngeq 14$ vertices. However, it was only conjectured by the authors about the minimum number of cycles of length 4 for maximal planar graphs with the remaining small vertex numbers. In this note we confirm their conjecture.