Abstract
Given a graph , the tree cover number of the graph, denoted , 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 -dimensional hypercube is 2 for all .
Citation
Chassidy Bozeman. Minerva Catral. Brendan Cook. Oscar González. Carolyn Reinhart. "On the tree cover number of a graph." Involve 10 (5) 767 - 779, 2017. https://doi.org/10.2140/involve.2017.10.767
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