Expanders are universal for the class of all spanning trees

Given a class of graphs ℱ, we say that a graph G is universal for ℱ, or ℱ-universal, if every H ∈ ℱ is contained in G as a subgraph. The construction of sparse universal graphs for various families ℱ has received a considerable amount of attention. One is particularly interested in tight ℱ-universal graphs, i.e., 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. Given integers n and Δ, we denote by script T(n, Δ) the class of all n-vertex trees with maximum degree at most Δ. In this work, we show that every n-vertex graph satisfying certain natural expansion properties is script T(n, Δ)-universal or, in other words, contains every spanning tree of maximum degree at most Δ. Our methods also apply to the case when Δ is some function of n. The result has a few very interesting implications. Most importantly, since random graphs are known to be good expanders, we obtain that the random graph G(n, p) is asymptotically almost surely (a.a.s.) universal for the class of all bounded degree spanning (that is, n-vertex) trees provided that p ≥ cn ^-1/3 log ² n where c > 0 is a constant. Moreover, a corresponding result holds for the random regular graph of degree pn. In fact, we show that if Δ satisfies log n ≤ Δ ≤ n ^1/3, then the random graph G(n, p) with p ≥ cΔn ^-1/3 log n and the random r-regular n-vertex graph with r ≥ cΔn ^2/3 log n are a.a.s. universal for script T(n, Δ). Another interesting consequence is the existence of locally sparse n-vertex graphs that are universal for script T(n, Δ). For Δ ∈ O(1), we show that one can (randomly) construct n-vertex script T(n, Δ)-universal graphs with clique number at most five. This complements the construction of Bhatt, Chung, Leighton, and Rosenberg (1989), whose script T(n, Δ)-universal graphs with merely O(n) edges contain large cliques of size Ω(Δ). We also derive some lower bounds and show that there exist very good expanders which are not universal for script T(n, Δ). In particular, we see that there are expanders of minimum degree Ω(n/log n) which are not script T(n, c√n)-universal. Finally, we show robustness of random graphs with respect to being universal for script T(n, Δ) in the context of the Maker-Breaker tree-universality game.