Home > RHUMJ > Vol. 12 (2011) > Iss. 2

#### Article Title

#### Abstract

The goal of this paper is to study the spanning trees of the 3-cube by understanding their \emph{edge slide graph}. A spanning tree of a graph $G$ is a minimal set of edges that connects all vertices. An edge slide occurs in a spanning tree of the 3-cube when a single edge can be slid across a 2-dimensional face to form another spanning tree. The edge slide graph is the graph whose vertices are the spanning trees, with an edge between two vertices if the spanning trees are related by a single edge slide. This report completely determines the edge slide graph of the 3-cube. The edge slide graph of the 3-cube has twelve components isomorphic to the 4-cube, and three other components, mutually isomorphic, with 64 vertices each. The main result is to determine the structure of the three components that each have 64 vertices and we also describe their symmetries. Some partial results on the 4-cube are also provided.

#### Sponsor

Chris Tuffley, Institute of Fundamental Sciences, Massey University Manawatu

#### Recommended Citation

Henden, Lyndal
(2011)
"The Edge Slide Graph of the 3-cube,"
*Rose-Hulman Undergraduate Mathematics Journal*: Vol. 12
:
Iss.
2
, Article 6.

Available at:
https://scholar.rose-hulman.edu/rhumj/vol12/iss2/6