Involve: A Journal of Mathematics
- Volume 10, Number 5 (2017), 767-779.
On the tree cover number of a graph
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 .
Involve, Volume 10, Number 5 (2017), 767-779.
Received: 13 November 2015
Revised: 7 September 2016
Accepted: 7 September 2016
First available in Project Euclid: 19 October 2017
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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
- Sets used in the proof of Theorem 10.