The degree-diameter problem for sparse graph classes

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 $Θ(Δ^{\floor{k/2}})$, 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.

References in corpus (2)

• Diameter and Treewidth in Minor-Closed Graph Families
• On the maximum order of graphs embedded in surfaces