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

Abstract

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