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

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Abstract

Given a graph G, the tree cover number of the graph, denoted T(G), 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 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


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References

  • F. Barioli, S. M. Fallat, L. H. Mitchell, and S. K. Narayan, “Minimum semidefinite rank of outerplanar graphs and the tree cover number”, Electron. J. Linear Algebra 22 (2011), 10–21.
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Supplemental materials

  • Sets used in the proof of Theorem 10.