Largest planar graphs of diameter $3$ and fixed maximum degree -- connection with fractional matchings

The degree diameter problem asks for the maximum possible number of vertices in a graph of maximum degree $\Delta$ and diameter $D$. In this paper, we focus on planar graphs of diameter $3$. Fellows, Hell and Seyffarth (1995) proved that for all $\Delta\geq 8$, the maximum number $\mathrm{np}_{\Delta, D}$ of vertices of a planar graph with maximum degree at most $\Delta$ and diameter at most 3 satisfies $\frac{9}{2}\Delta - 3 \leq \mathrm{np}_{\Delta,3} \leq 8 \Delta + 12$. We show that the lower bound they gave is optimal, up to an additive constant, by proving that there exists $c>0$ such that $\mathrm{np}_{\Delta,3} \leq \frac{9}{2}\Delta + c$ for every $\Delta\geq 0$. Our proof consists in a reduction to the fractional maximum matching problem on a specific class of planar graphs, for which we show that the optimal solution is $\tfrac{9}{2}$, and characterize all graphs attaining this bound.