Full Degree Spanning Trees in Random Cubic Graphs

We study the problem of maximizing the number of full degree vertices in a spanning tree T of a graph G; that is, the number of vertices whose degree in T equals its degree in G. In cubic graphs, this problem is equivalent to maximizing the number of leaves in T and minimizing the size of a connected dominating set of G. We provide an algorithm that, with high probability, produces a tree with at least 0.437n vertices of full degree when run on a random cubic graph. This improves the previously best known lower bound of 0.4146n.