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

TL;DR

This paper establishes tight bounds on the maximum size of planar graphs with diameter 3 and bounded degree, linking the problem to fractional matchings and characterizing extremal graphs.

Contribution

It proves the lower bound is essentially optimal by connecting the degree diameter problem to fractional maximum matchings on planar graphs.

Findings

The lower bound for the number of vertices is tight up to an additive constant.

Optimal fractional maximum matchings have value 4.5 on certain planar graphs.

Characterization of graphs attaining the maximum fractional matching bound.

Abstract

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.