Universal Graphs for Bounded-Degree Trees and Planar Graphs

Abstract

How small can a graph be that contains as subgraphs all trees on n vertices with maximum degree d? In this paper, this question is answered by constructing such universal graphs that have n vertices and bounded degree (depending only on d). Universal graphs with n vertices and $O(n\log n)$ edges are also constructed that contain all bounded-degree planar graphs on n vertices as subgraphs. In general, it is shown that the minimum universal graph containing all bounded-degree graphs on n vertices with separators of size $n^\alpha $ has $O(n)$ edges if $\alpha < \frac{1}{2}$; $O(n\log n)$ edges if $\alpha = \frac{1}{2}$; $O(n^{2\alpha } )$ edges if $\alpha > \frac{1}{2}$.