The degree-diameter problem for sparse graph classes

TL;DR

This paper investigates the maximum size of graphs with given degree and diameter constraints within various sparse graph classes, providing tight bounds and specific results for classes like bounded average degree, arboricity, treewidth, and surface-embeddable graphs.

Contribution

It establishes new bounds for the degree-diameter problem tailored to specific sparse graph classes, extending classical results to more complex graph families.

Findings

For graphs of bounded average degree, maximum vertices are Θ(Δ^{k-1}).

For graphs of bounded arboricity, maximum vertices are Θ(Δ^{⌊k/2⌋}).

Precise bounds are provided for graphs with given treewidth, surface embedding, and apex-minor-free properties.

Abstract

The degree-diameter problem asks for the maximum number of vertices in a graph with maximum degree $\Delta$ and diameter $k$. For fixed $k$, the answer is $\Theta(\Delta^k)$. We consider the degree-diameter problem for particular classes of sparse graphs, and establish the following results. For graphs of bounded average degree the answer is $\Theta(\Delta^{k-1})$, and for graphs of bounded arboricity the answer is $\Theta(\Delta^{\floor{k/2}})$, in both cases for fixed $k$. For graphs of given treewidth, we determine the the maximum number of vertices up to a constant factor. More precise bounds are given for graphs of given treewidth, graphs embeddable on a given surface, and apex-minor-free graphs.