Full Degree Spanning Trees in Random Regular Graphs

TL;DR

This paper investigates maximizing full degree vertices in spanning trees of random regular graphs, introducing an improved algorithm that guarantees a higher number of such vertices than previous methods.

Contribution

The authors present a new algorithm that achieves a higher lower bound on full degree vertices in spanning trees of random regular graphs, improving previous results.

Findings

Achieves at least 0.4591n full degree vertices in random cubic graphs.

Provides lower bounds for full degree vertices in random regular graphs for r ≤ 10.

Improves the previous lower bound of 0.4146n for cubic graphs.

Abstract

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 which produces (w.h.p.) a tree with at least $0.4591n$ vertices of full degree (and also, leaves) when run on a random cubic graph. This improves the previously best known lower bound of $0.4146 n$. We also provide lower bounds on the number of full degree vertices in the random regular graph $G(n,r)$ for $r \le 10$.