On the star arboricity of hypercubes

Abstract

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.