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arXiv · 1307.4456

The degree-diameter problem for sparse graph classes

Guillermo Pineda-Villavicencio·David R. Wood
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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.

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Guillermo Pineda-Villavicencio, David R. Wood. 2013-07-17. The degree-diameter problem for sparse graph classes. https://arxiv.org/abs/1307.4456

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