On the tree cover number of a graph

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 $d\ge 2$.