# Spanning Trees and Domination in Hypercubes

## TL;DR

This paper investigates the properties of spanning trees and dominating sets in hypercube graphs, revealing asymptotic behaviors and constructing specific trees using coding theory and expansion techniques.

## Contribution

It provides new asymptotic estimates for the maximum leaves in spanning trees and the minimum connected dominating sets in hypercubes, employing Hamming codes and expansion methods.

## Key findings

- Maximum leaves in spanning trees of hypercubes grow exponentially with n.
- Minimum connected dominating set size is approximately 2^n/n.
- Constructive methods for leafy spanning trees using coding theory.

## Abstract

Let $L(G)$ denote the maximum number of leaves in any spanning tree of a connected graph $G$. We show the (known) result that for the $n$-cube $Q_n$, $L(Q_n) \sim 2^n = |V(Q_n)|$ as $n\rightarrow \infty$. Examining this more carefully, consider the minimum size of a connected dominating set of vertices $\gamma_c(Q_n)$, which is $2^n-L(Q_n)$ for $n\ge2$. We show that $\gamma_c(Q_n)\sim 2^n/n$. We use Hamming codes and an "expansion" method to construct leafy spanning trees in $Q_n$.