The degree-diameter problem consists of finding the maximum number of vertices $n$ of a graph with diameter $d$ and maximum degree $\Delta$. 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 $\Delta \geq 8$, then $n \leq 3\Delta - 1$ for such graphs. This bound is sharp for $\Delta$ odd.

翻译：度直径问题是指寻找图中直径 $d$ 和最大度 $\Delta$ 条件下顶点数 $n$ 的最大值。该问题已被广泛研究，并在每个面均由3-圈（三角剖分）界定的低直径平面图、以及每个面均由4-圈（四边形剖分）界定的平面图中得到解决。本文解决了每个面均由5-圈（五边形剖分）界定的直径3平面图的度直径问题。我们证明：若 $\Delta \geq 8$，则此类图的顶点数满足 $n \leq 3\Delta - 1$。该上界在 $\Delta$ 为奇数时是紧确的。