Perfect matchings in planar 3-connected graphs: girth five and minimum degree five

Let $\mathrm{\Phi }(G)$ denote the number of perfect matchings of a graph $G$. Goedgebeur et al. posted two questions about the perfect matchings of planar 3-connected graphs: does there exist an infinite family of matchable planar 3-connected graphs of girth 5 with a constant number of perfect matchings? Whether there exist infinitely many planar 3-connected graphs with minimum degree 5 which have a constant number of perfect matchings. We present answers to those questions. First, we construct, for every even $n\ge 196$, a simple planar 3-connected graph of order $n$ and girth five with exactly $c$ perfect matchings, where $c$ is a fixed positive integer. Second, for a simple planar 3-connected graph $G$ with a perfect matching and minimum degree at least five, we show that $\mathrm{\Phi }(G)\ge 2^{\lceil (\mathrm{\mid }V(G)\mathrm{\mid }+20)\mathrm{/}22\rceil }$. This implies that, for each positive integer $t$, only finitely many such graphs, up to isomorphism, have at most $t$ perfect matchings.