On the star arboricity of hypercubes

A hypercube Qn 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, 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 sa(Qn) for n ∈ {2k − 3, 2 + 1, 2 + 2, 2 + 2 − 4}, i ≥ j ≥ 2. We also improve the last known upper bound of sa(Qn) and show the relation between sa(G) and square coloring.