On the star arboricity of hypercubes

A Hypercube \(Q_n\) is a graph in which the vertices are all binary vectors of length n, and two vertices are adjacent if and only if their components differ in exactly one place. A galaxy or a star forest is a union of vertex disjoint stars. The star arboricity of a graph \(G\), \({\rm sa}(G)\), is...

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Bibliographic Details
Published inarXiv.org
Main Authors Karisani, Negin, Mahmoodian, E S, Sobhani, Narges K
Format Paper
LanguageEnglish
Published Ithaca Cornell University Library, arXiv.org 29.06.2014
Subjects
Apexes
Codes
Coloring
Galaxies
Graph theory
Hypercubes
Upper bounds
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Summary:A Hypercube \(Q_n\) is a graph in which the vertices are all binary vectors of length n, and two vertices are adjacent if and only if their components differ in exactly one place. A galaxy or a star forest is a union of vertex disjoint stars. The star arboricity of a graph \(G\), \({\rm sa}(G)\), is the minimum number of galaxies which partition the edge set of \(G\). In this paper among other results, we determine the exact values of \({\rm sa}(Q_n)\) for \(n \in \{2^k-3, 2^k+1, 2^k+2, 2^i+2^j-4\}\), \(i \geq j \geq 2\). We also improve the last known upper bound of \({\rm sa}(Q_n)\) and show the relation between \({\rm sa}(G)\) and square coloring.
ISSN:2331-8422