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 ∆, we denote by T (n, ∆) the class of all n-vertex trees with maximum degree at most ∆. In this work, we show that every n-vertex graph satisfying certain natural expansion properties is T (n, ∆)-universal or, in other words, contains every spanning tree of maximum degree at most ∆. Our methods also apply to the case when ∆ 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 th...

• Type: Journal
• Year: 2012
• Where: SODA
• Authors: Daniel Johannsen, Michael Krivelevich, Wojciech Samotij
• Keywords: Algorithms, Random Graph, SODA 2012, Universal Graphs, Vertex Graph