## Involve: A Journal of Mathematics

- Involve
- Volume 10, Number 5 (2017), 767-779.

### On the tree cover number of a graph

Chassidy Bozeman, Minerva Catral, Brendan Cook, Oscar González, and Carolyn Reinhart

#### Abstract

Given a graph $G$, the tree cover number of the graph, denoted $T\left(G\right)$, is the minimum number of vertex disjoint simple trees occurring as induced subgraphs that cover all the vertices of G. This graph parameter was introduced in 2011 as a tool for studying the maximum positive semidefinite nullity of a graph, and little is known about it. It is conjectured that the tree cover number of a graph is at most the maximum positive semidefinite nullity of the graph.

In this paper, we establish bounds on the tree cover number of a graph, characterize when an edge is required to be in some tree of a minimum tree cover, and show that the tree cover number of the $d$-dimensional hypercube is 2 for all $d\ge 2$.

#### Article information

**Source**

Involve, Volume 10, Number 5 (2017), 767-779.

**Dates**

Received: 13 November 2015

Revised: 7 September 2016

Accepted: 7 September 2016

First available in Project Euclid: 19 October 2017

**Permanent link to this document**

https://projecteuclid.org/euclid.involve/1508433091

**Digital Object Identifier**

doi:10.2140/involve.2017.10.767

**Mathematical Reviews number (MathSciNet)**

MR3652446

**Zentralblatt MATH identifier**

1364.05022

**Subjects**

Primary: 05C05: Trees 05C50: Graphs and linear algebra (matrices, eigenvalues, etc.) 05C76: Graph operations (line graphs, products, etc.)

**Keywords**

tree cover number hypercube maximum nullity minimum rank

#### Citation

Bozeman, Chassidy; Catral, Minerva; Cook, Brendan; González, Oscar; Reinhart, Carolyn. On the tree cover number of a graph. Involve 10 (2017), no. 5, 767--779. doi:10.2140/involve.2017.10.767. https://projecteuclid.org/euclid.involve/1508433091

#### Supplemental materials

- Sets used in the proof of Theorem 10. Supplemental files are immediately available to subscribers. Non-subscribers gain access to supplemental files with the purchase of the article.