#### Vol. 10, No. 5, 2017

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On the tree cover number of a graph

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

Vol. 10 (2017), No. 5, 767–779
##### 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 $d\ge 2$.

##### Keywords
tree cover number, hypercube, maximum nullity, minimum rank
##### Mathematical Subject Classification 2010
Primary: 05C05, 05C50, 05C76
##### Supplementary material

Sets used in the proof of Theorem 10