The degree-diameter problem for plane graphs with pentagonal faces

The degree-diameter problem consists of finding the maximum number of vertices $n$ of a graph with diameter $d$ and maximum degree $\mathrm{\Delta }$. This problem is well studied, and has been solved for plane graphs of low diameter in which every face is bounded by a 3-cycle (triangulations), and plane graphs in which every face is bounded by a 4-cycle (quadrangulations). In this paper, we solve the degree diameter problem for plane graphs of diameter 3 in which every face is bounded by a 5-cycle (pentagulations). We prove that if $\mathrm{\Delta }\ge 8$, then $n\le 3\mathrm{\Delta }-1$ for such graphs. This bound is sharp for $\mathrm{\Delta }$ odd.