Expanders are universal for the class of all spanning trees

A graph is called universal for a given graph class ℱ (or, equivalently, ℱ-universal) if it contains a copy of every graph in ℱ as a subgraph. The construction of sparse universal graphs for various classes ℱ has received a considerable amount of attention. There is particular interest in tight ℱ-universal graphs, that is, graphs whose number of vertices is equal to the largest number of vertices in a graph from ℱ. Arguably, the most studied case is that when ℱ is some class of trees. In this work, we are interested in Τ(n,Δ), the class of all n-vertex trees with maximum degree at most Δ. We show that every n-vertex graph satisfying certain natural expansion properties is Τ(n,Δ)-universal. Our methods also apply to the case when Δ is some function of n. Since random graphs are known to be good expanders, our result implies, in particular, that there exists a positive constant c such that the random graph G(n,cn ^-1/3log² n) is asymptotically almost surely (a.a.s.) universal for Τ(n,O(1)). Moreover, a corresponding result holds for the random regular graph of degree cn^2/3log² n. Another interesting consequence is the existence of locally sparse n-vertex Τ(n,Δ)-universal graphs. For example, we show that one can (randomly) construct n-vertex Τ(n,O(1))-universal graphs with clique number at most five. This complements the construction of Bhatt, Chung, Leighton and Rosenberg (1989), whose Τ(n,Δ)-universal graphs with merely O(n) edges contain large cliques of size Ω(Δ). Finally, we show that random graphs are robustly Τ(n,Δ)-universal in the context of the Maker-Breaker tree-universality game.